Angle

<h2 id="fundamentals">Fundamentals</h2>
An angle is a figure lying in a plane formed by two distinct rays (half-lines emanating indefinitely from an endpoint in one direction), which share a common endpoint. The rays are called the sides or arms of the angle, and the common endpoint is called the vertex. The sides divide the plane into two regions: the interior of the angle and the exterior of the angle.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h3>Notation</h3>

An angle symbol (
 
 
 
 ∠
 
 
 {\displaystyle \angle }
 
 or 
 
 
 
 
 
 
 
 ^
 
 
 
 
 
 {\displaystyle {\widehat {\quad }}}
 
, read as "angle") together with one or three defining points is used to identify angles in geometric figures. For example, the angle with vertex A formed by the <a href="/facts/Ray_(geometry)/gJqsr4Gj">rays</a> 
 
 
 
 
 
 
 
 A
 B
 
 →
 
 
 
 
 
 {\displaystyle {\vec {AB}}}
 
 and 
 
 
 
 
 
 
 AC
 →
 
 
 
 
 
 {\displaystyle {\vec {\text{AC}}}}
 
 is denoted as 
 
 
 
 ∠
 A
 
 
 {\displaystyle \angle A}
 
 (using the vertex alone) or 
 
 
 
 ∠
 
 BAC
 
 
 
 {\displaystyle \angle {\text{BAC}}}
 
 (with the vertex always named in the middle). The size or measure of the angle is denoted 
 
 
 
 m
 ∠
 
 A
 
 
 
 {\displaystyle m\angle {\text{A}}}
 
 or 
 
 
 
 m
 ∠
 
 BAC
 
 
 
 {\displaystyle m\angle {\text{BAC}}}
 
.
In geometric figures and <a href="/facts/Expression_(mathematics)/MPrMlYbE">mathematical expressions</a>, it is also common to use <a href="/facts/Greek_letter/1cHwB8Dg">Greek letters</a> (α, β, γ, θ, φ, ...) or lower case Roman letters (a, b, c, ...) as <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a> to represent the size of an angle.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The Greek letter <a href="/facts/Pi_(letter)/4UpopgBE">π</a> is typically not used for this purpose to avoid confusion with the <a href="/facts/Pi/vC0UBj1q">circle constant</a>.
Conventionally, angle size is measured "between" the sides through the interior of the angle and given as a <a href="/facts/Magnitude_(mathematics)/KCn9PoF0">magnitude</a> or <a href="/facts/Scalar_(physics)/05uvvP8N">scalar quantity</a>. At other times it might be measured through the exterior of the angle or given as a <a href="/facts/Sign_(mathematics)/z0xollg1">signed number</a> to indicate a direction of measurement (see § Signed angles).

<h3>Units of measurement</h3>
Angles are measured in various units, the most common being the <a href="/facts/Degree_(angle)/f48Ns8SX">degree</a> (denoted by the symbol °), <a href="/facts/Radian/Srd38Sxa">radian</a> (denoted by the symbol rad) and <a href="/facts/Turn_(angle)/QczxAyuM">turn</a>. These units differ in the way they divide up a full angle, an angle where one ray, initially congruent to the other, performs a compete rotation about the vertex to return back to its starting position.
Degrees and turns are defined directly with reference to a full angle, which measures 1 turn or 360°. A measure in turns gives an angle's size as a proportion of a full angle and a degree can be considered as a subdivision of a turn. Radians are not defined directly in relation to a full angle (see § Measuring angles), but in such a way that its measure is 2π rad, approximately 6.28 rad.

<h3>Common angles</h3>
<ul><li>An angle equal to 0° or not turned is called a zero angle.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>An angle smaller than a right angle (less than 90°) is called an acute angle<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> ("acute" meaning "sharp").</li>
<li>An angle equal to ⁠1/4⁠ <a href="/facts/Turn_(angle)/QczxAyuM">turn</a> (90° or ⁠π/2⁠ <a href="/facts/Radian/Srd38Sxa">radians</a>) is called a <a href="/facts/Right_angle/psaslRq7">right angle</a>. Two lines that form a right angle are said to be <a href="/facts/Normal_(geometry)/jhtLIhcu">normal</a>, <a href="/facts/Orthogonality/JVIjYPog">orthogonal</a>, or <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> ("obtuse" meaning "blunt").</li>
<li>An angle equal to ⁠1/2⁠ turn (180° or π radians) is called a straight angle.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li>
<li>An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle.</li>
<li>An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, round angle or perigon.</li>
<li>An angle that is not a multiple of a right angle is called an oblique angle.</li></ul>
The names, intervals, and measuring units are shown in the table below:

<table><tbody><tr><td>Name  </td><td>zero angle</td><td>acute angle</td><td>right angle</td><td>obtuse angle</td><td>straight angle</td><td>reflex angle</td><td>full angle</td></tr><tr><th>Unit</th><th colspan="10"><a href="/facts/Interval_(mathematics)/e9se3vTe">Interval</a></th></tr><tr><td><a href="/facts/Turn_(geometry)/QczxAyuM">turn</a>  </td><td>0 turn</td><td>(0, ⁠1/4⁠) turn</td><td>⁠1/4⁠ turn</td><td>(⁠1/4⁠, ⁠1/2⁠) turn</td><td>⁠1/2⁠ turn</td><td>(⁠1/2⁠, 1) turn</td><td>1 turn</td></tr><tr><td><a href="/facts/Degree_(angle)/f48Ns8SX">degree</a>  </td><td>0°</td><td>(0, 90)°</td><td>90°</td><td>(90, 180)°</td><td>180°</td><td>(180, 360)°</td><td>360°</td></tr><tr><td><a href="/facts/Radian/Srd38Sxa">radian</a></td><td>0 rad</td><td>(0, ⁠1/2⁠π) rad</td><td>⁠1/2⁠π rad</td><td>(⁠1/2⁠π, π) rad</td><td>π rad</td><td>(π, 2π) rad</td><td>2π rad</td></tr></tbody></table>
<h3>Addition and subtraction</h3>

The angle addition postulate states that if D is a point lying in the interior of 
 
 
 
 ∠
 
 BAC
 
 
 
 {\displaystyle \angle {\text{BAC}}}
 
 then:<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> 
 
 
 
 m
 ∠
 
 BAC
 
 =
 m
 ∠
 
 BAD
 
 +
 m
 ∠
 
 DAC
 
 .
 
 
 {\displaystyle m\angle {\text{BAC}}=m\angle {\text{BAD}}+m\angle {\text{DAC}}.}
 
 This relationship defines what it means add any two angles: their vertices are placed together while sharing a side to create a new larger angle. The measure of the new larger angle is the sum of the measures of the two angles. Subtraction follows from rearrangement of the formula.

<h2 id="types">Types</h2>
"Oblique angle" redirects here. For the cinematographic technique, see <a href="/facts/Dutch_angle/eEcdA6vH">Dutch angle</a>.
<h3>Adjacent and vertical angles</h3>

"Vertical angle" redirects here and is not to be confused with <a href="/facts/Zenith_angle/tcm21e6t">Zenith angle</a>.
Adjacent angles (abbreviated adj. ∠s), are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called complementary, supplementary, and explementary angles (see § Combining angle pairs below).

Vertical angles are formed when two straight lines intersect at a point producing four angles. A pair of angles opposite each other are called vertical angles, opposite angles or vertically opposite angles (abbreviated vert. opp. ∠s),<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> where "vertical" refers to the sharing of a vertex, rather than an up-down orientation. The vertical angle theorem states that vertical angles are always congruent or equal to each other.
A <a href="/facts/Transversal_(geometry)/uoI7mdNj">transversal</a> is a line that intersects a pair of (often parallel) lines and is associated with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h3>Combining angle pairs</h3>
When summing two angles that are either adjacent or separated in space, three cases are of particular importance.

<h4>Complementary angles</h4>

Complementary angles are angle pairs whose measures sum to a right angle (⁠1/4⁠ turn, 90°, or ⁠π/2⁠ rad).<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> If the two complementary angles are adjacent, their non-shared sides form a right angle. In a <a href="/facts/Right_triangle/dD6M9GAn">right-angle triangle</a> the two acute angles are complementary as the sum of the internal angles of a <a href="/facts/Triangle/cRXUwsMn">triangle</a> is 180°.
The difference between an angle and a right angle is termed the complement of the angle<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> which is from the Latin complementum and associated verb complere, meaning "to fill up". An acute angle is "filled up" by its complement to form a right angle.
Complementary angles are fundamental to <a href="/facts/Trigonometry/K4hfaC6F">trigonometry</a> and the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a>.

<h4>Supplementary angles</h4>

Two angles that sum to a straight angle (⁠1/2⁠ turn, 180°, or π rad) are called supplementary angles.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> If the two supplementary angles are adjacent, their non-shared sides form a straight angle or <a href="/facts/Line_(geometry)/gJqsr4Gj">straight line</a> and are called a linear pair of angles.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> The difference between an angle and a straight angle is termed the supplement of the angle.
Examples of non-adjacent complementary angles include the consecutive angles a <a href="/facts/Parallelogram/DRuktq60">parallelogram</a> and opposite angles of a <a href="/facts/Cyclic_quadrilateral/FNwQ0cda">cyclic quadrilateral</a>. For a circle with center O, and <a href="/facts/Tangent_lines_to_circles/jlj3WjKK">tangent lines</a> from an exterior point P touching the circle at points T and Q, the resulting angles ∠TPQ and ∠TOQ are supplementary.

<h4>Explementary angles</h4>

Two angles that sum to a full angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> The difference between an angle and a full angle is termed the explement or conjugate of the angle.

<h3>Polygon-related angles</h3>

<ul><li>An angle that is part of a <a href="/facts/Simple_polygon/C5CfrPWf">simple polygon</a> is called an <a href="/facts/Interior_angle/jd78Qkbt">interior angle</a> if it lies on the inside of that simple polygon. A simple <a href="/facts/Concave_polygon/opSnBwgh">concave polygon</a> has at least one interior angle, that is, a reflex angle. In <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>, the measures of the interior angles of a <a href="/facts/Triangle/cRXUwsMn">triangle</a> add up to π radians, 180°, or ⁠1/2⁠ turn; the measures of the interior angles of a simple <a href="/facts/Convex_polygon/7x35w610">convex</a> <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex <a href="/facts/Polygon/xvkmYhvG">polygon</a> with n sides add up to (n − 2)π radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)⁠1/2⁠ turn.</li>
<li>The supplement of an interior angle is called an <a href="/facts/Exterior_angle/jd78Qkbt">exterior angle</a>; that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> If the corresponding interior angle is a reflex angle, the exterior angle should be considered <a href="/facts/Negative_number/LzErzCmI">negative</a>. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an <a href="/facts/Orientation_(space)/1SEQM2Pt">orientation</a> of the <a href="/facts/Plane_(mathematics)/tAUtRFcn">plane</a> (or <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a>) to decide the sign of the exterior angle measure. In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in <a href="/facts/Logo_(programming_language)/Wh4sdwAh">Logo Turtle programs</a> when drawing regular polygons.</li>
<li>In a <a href="/facts/Triangle/cRXUwsMn">triangle</a>, the <a href="/facts/Bisection/JTsya7Ca">bisectors</a> of two exterior angles and the bisector of the other interior angle are <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent</a> (meet at a single point).<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a>: 149 </li>
<li>In a triangle, three intersection points, each of an external angle bisector with the opposite <a href="/facts/Extended_side/wtqKRrBS">extended side</a>, are <a href="/facts/Collinearity/SxbVBJYR">collinear</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>: 149 </li>
<li>In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>: 149 </li>
<li>Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> This conflicts with the above usage.</li></ul>
<h3>Plane-related angles</h3>
<ul><li>The angle between two <a href="/facts/Plane_(mathematics)/tAUtRFcn">planes</a> (such as two adjacent faces of a <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a>) is called a <a href="/facts/Dihedral_angle/C9WEJsuS">dihedral angle</a>.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> It may be defined as the acute angle between two lines <a href="/facts/Normal_(geometry)/jhtLIhcu">normal</a> to the planes.</li>
<li>The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the <a href="/facts/Normal_(geometry)/jhtLIhcu">normal</a> to the plane.</li></ul>
<h2 id="measuring-angles">Measuring angles </h2>
See also: <a href="/facts/Angle_measuring_instrument/DoKGJII2">Angle measuring instrument</a>

Measurement of angles is intrinsically linked with circles and rotation. An angle is measured by placing it within a circle of any size, with the vertex at the circle's centre and the sides intersecting the perimeter. 
An <a href="/facts/Circular_arc/RrehhCQR">arc</a> s is formed as the shortest distance on the perimeter between the two points of intersection, which is said to be the arc <a href="/facts/Subtended_angle/GQxln7lP">subtended</a> by the angle. 
The length of s can be used to measure the angle's size 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
, however as s is dependent on the size of the circle chosen, it must be adjusted so that any arbitrary circle will give the same measure of angle. This can be done in two ways: by taking the ratio to either the radius r or circumference C of the circle.
The ratio of the length s by the radius r is the number of <a href="/facts/Radian/Srd38Sxa">radians</a> in the angle, while the ratio of length s by the circumference C is the number of <a href="/facts/Turn_(angle)/QczxAyuM">turns</a>:<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>

θ
          
            
              r
              a
              d
            
          
        
        =
        
          
            s
            r
          
        
        
        
          r
          a
          d
        
        
        
        
          θ
          
            
              t
              u
              r
              n
            
          
        
        =
        
          
            s
            C
          
        
         
        =
        
          
            s
            
              2
              π
              r
            
          
        
        
        
          t
          u
          r
          n
          s
        
      
    
    {\displaystyle \theta _{\mathrm {rad} }={\frac {s}{r}}\,\mathrm {rad} \qquad \qquad \theta _{\mathrm {turn} }={\frac {s}{C}}\ ={\frac {s}{2\pi r}}\,\mathrm {turns} }

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratios ⁠s/r⁠ and ⁠s/C⁠ are unaltered.<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>
Angles of the same size are said to be equal congruent or equal in measure.

<h3>Units</h3>
In addition to the radian and turn, other angular units exist, typically based on subdivisions of the turn, including the <a href="/facts/Degree_(angle)/f48Ns8SX">degree</a> (°) and the <a href="/facts/Gradian/FoTX1r68">gradian</a> (grad), though many others have been used throughout <a href="/facts/History_of_Mathematics/18h5rQA5">history</a>.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a>
Conversion between units may be obtained by multiplying the angular measure in one unit by a conversion constant of the form 
 
 
 
 
 
 
 
 k
 
 a
 
 
 
 k
 
 b
 
 
 
 
 
 
 
 {\displaystyle {\tfrac {k_{a}}{k_{b}}}}
 
 where 
 
 
 
 
 
 k
 
 a
 
 
 
 
 
 {\displaystyle {k_{a}}}
 
 and 
 
 
 
 
 
 k
 
 b
 
 
 
 
 
 {\displaystyle {k_{b}}}
 
 are the measures of a complete turn in units a and b. For example, to convert an angle of 
 
 
 
 
 
 
 π
 2
 
 
 
 
 
 {\displaystyle {\tfrac {\pi }{2}}}
 
radians to degrees:

θ
          
            deg
          
        
        =
        
          
            
              k
              
                deg
              
            
            
              k
              
                
                  r
                  a
                  d
                
              
            
          
        
        ⋅
        
          θ
          
            
              r
              a
              d
            
          
        
        =
        
          
            
              360
              
                ∘
              
            
            
              2
              π
              
              
                r
                a
                d
              
            
          
        
        ⋅
        
          
            π
            2
          
        
        
        
          r
          a
          d
        
        =
        
          90
          
            ∘
          
        
      
    
    {\displaystyle \theta _{\deg }={\frac {k_{\deg }}{k_{\mathrm {rad} }}}\cdot \theta _{\mathrm {rad} }={\frac {360^{\circ }}{2\pi \,\mathrm {rad} }}\cdot {\frac {\pi }{2}}\,\mathrm {rad} =90^{\circ }}

The following table lists some units used to represent angles.

<table><tbody><tr><th>Name (symbol)</th><th>Number in one turn</th><th>1 unit in degrees</th><th>Description</th></tr><tr><td><a href="/facts/Turn_(geometry)/QczxAyuM">turn</a></td><td>1</td><td>360°</td><td>The turn is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2π or <a href="/facts/Tau_(mathematics)/gfvnmJoH">𝜏</a> radians.</td></tr><tr><td><a href="/facts/Degree_(angle)/f48Ns8SX">degree</a> ( ° )</td><td>360</td><td>1°</td><td>The degree is a <a href="/facts/Sexagesimal/bspNRWdR">sexagesimal</a> subunit of the sextant, making one turn equal to 360°.</td></tr><tr><td><a href="/facts/Radian/Srd38Sxa">radian</a> (rad)</td><td>2π</td><td>57.2957...°</td><td>The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius.</td></tr><tr><td><a href="/facts/Grad_(angle)/FoTX1r68">grad</a> (gon)</td><td>400</td><td>0.9°</td><td>The grad, also called grade, <a href="/facts/Gradian/FoTX1r68">gradian</a>, or gon, is defined as ⁠1/100⁠ of a right angle. The grad is used mostly in <a href="/facts/Triangulation_(surveying)/XDQvlPzl">triangulation</a> and continental <a href="/facts/Surveying/qchGcnRC">surveying</a>.</td></tr><tr><td><a href="/facts/Arcminute/tybs6YGC">arcminute</a> ( ′ )</td><td>21,600</td><td>⁠1/60⁠°</td><td>The minute of arc (or arcminute, or just minute) is a sexagesimal subunit of a degree.</td></tr><tr><td><a href="/facts/Arcsecond/tybs6YGC">arcsecond</a> ( ″ )</td><td>1,296,000</td><td>⁠1/3600⁠°</td><td>The second of arc (or arcsecond, or just second) is a sexagesimal subunit of a minute of arc.</td></tr><tr><td><a href="/facts/Milliradian/vCn6rKUH">milliradian</a> (mrad)</td><td>2000π</td><td>~0.0573°</td><td>The milliradian is a thousandth of a radian. For artillery and navigation a unit is used, often called a 'mil', which are approximately equal to a milliradian. One turn is exactly 6000, 6300, or 6400 mils, depending on which definition is used.</td></tr><tr><td><a href="/facts/Hour_angle/rj9KG0Sz">hour angle</a></td><td>24</td><td>15°</td><td>The astronomical hour angle is ⁠1/24⁠ turn.</td></tr><tr><td><a href="/facts/Points_of_the_compass/jPKnTUYz">(compass) point</a></td><td>32</td><td>11.25°</td><td>The point or wind, used in <a href="/facts/Navigation/sGGAJ0AQ">navigation</a>, divides the compass (one turn) into 32 points or compass directions.</td></tr><tr><td><a href="/facts/Binary_angular_measurement/qHGoI2dt">binary degree</a></td><td>256</td><td>1.40625°</td><td>The binary degree, also known as the <a href="/facts/Binary_radian/qHGoI2dt">binary radian</a> or brad or binary angular measurement (BAM).<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a></td></tr><tr><td><a href="/facts/Circular_sector/TqwCaEA5">quadrant</a></td><td>4</td><td>90°</td><td>One quadrant is a ⁠1/4⁠ turn and also known as a <a href="/facts/Right_angle/psaslRq7">right angle</a>. In German, the symbol ∟ has been used to denote a quadrant.</td></tr><tr><td><a href="/facts/Circular_sector/TqwCaEA5">sextant</a></td><td>6</td><td>60°</td><td>The sextant was the unit used by the <a href="/facts/Babylonians/J4jjdUua">Babylonians</a>.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a><a class="footnote-ref" id="fnref:32" href="#fn:32">32</a></td></tr><tr><td>hexacontade</td><td>60</td><td>6°</td><td>The hexacontade is a unit used by <a href="/facts/Eratosthenes/ay7Pyfsa">Eratosthenes</a>, with 60 hexacontades in a turn.</td></tr><tr><td>pechus</td><td>144 to 180</td><td>2° to 2.5°</td><td>The pechus was a <a href="/facts/Babylonian_mathematics/9lN3d6ML">Babylonian</a> unit.</td></tr><tr><td>diameter part</td><td>~376.991</td><td>~0.95493°</td><td>The diameter part (occasionally used in Islamic mathematics) is ⁠1/60⁠ radian.</td></tr><tr><td>zam</td><td>224</td><td>~1.607°</td><td>In old Arabia, a <a href="/facts/Turn_(geometry)/QczxAyuM">turn</a> was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam.</td></tr></tbody></table>
<h3>Dimensional analysis</h3>
Further information: <a href="/facts/Radian/Srd38Sxa">Radian § Dimensional analysis</a>
In mathematics and the <a href="/facts/International_System_of_Quantities/MJFfvYNx">International System of Quantities</a>, an angle is defined as a dimensionless quantity, and in particular, the <a href="/facts/Radian/Srd38Sxa">radian</a> is defined as dimensionless in the <a href="/facts/International_System_of_Units/S7YLglNz">International System of Units</a>. This convention prevents angles providing information for <a href="/facts/Dimensional_analysis/t6UskoCD">dimensional analysis</a>.
While mathematically convenient, this has led to significant discussion among scientists and teachers. Some scientists have suggested treating the angle as having its own dimension, similar to length or time. This would mean that angle units like radians would always be explicitly present in calculations, making the dimensional analysis more straightforward. However, this approach would also require changing many well-known mathematical and physics formulas.

<h3>Signed angles</h3>
Main article: <a href="/facts/Angle_of_rotation/a4VjkXGe">Angle of rotation</a>
See also: <a href="/facts/Sign_(mathematics)/z0xollg1">Sign (mathematics) § Angles</a>, and <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space § Angle</a>

An angle denoted as ∠BAC might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, It is therefore frequently helpful to impose a convention that allows positive and negative angular values to represent <a href="/facts/Orientation_(geometry)/Tj24mTJ8">orientations</a> and/or <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotations</a> in opposite directions or "sense" relative to some reference.
In a two-dimensional <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinate system</a>, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive <a href="/facts/X-axis/lkTqvMmB">x-axis</a>, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive <a href="/facts/Y-axis/lkTqvMmB">y-axis</a> and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are <a href="/facts/Anticlockwise/3Npe4Swe">anticlockwise</a>, and negative cycles are <a href="/facts/Clockwise/3Npe4Swe">clockwise</a>.
In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).
In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an <a href="/facts/Orientability/1SEQM2Pt">orientation</a>, which is typically determined by a <a href="/facts/Normal_(geometry)/jhtLIhcu">normal vector</a> passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
In <a href="/facts/Navigation/sGGAJ0AQ">navigation</a>, <a href="/facts/Bearing_(navigation)/rE84IorW">bearings</a> or <a href="/facts/Azimuth/kEyt9nJ3">azimuth</a> are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

<h3>Equivalent angles</h3>
<ul><li>Angles that have the same measure (i.e., the same magnitude) are said to be equal or <a href="/facts/Congruence_(geometry)/2tKJ6AhB">congruent</a>. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all right angles are equal in measure).</li>
<li>Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.</li>
<li>The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a><a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude <a href="/facts/Modulo/5tKQmM76">modulo</a> ⁠1/2⁠ turn, 180°, or π radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).</li></ul>
<h3>Related quantities</h3>
For an angular unit, it is definitional that the <a href="/facts/Angle_addition_postulate/gFUTEQYq">angle addition postulate</a> holds, however some measurements or quantities related to angles are in use that do not satisfy this postulate:

<ul><li>The <a href="/facts/Slope/a2L3k0j5">slope</a> or gradient is equal to the <a href="/facts/Tangent_(trigonometric_function)/czej7ur0">tangent</a> of the angle and is often expressed as a percentage ("rise" over "run"). For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction. An <a href="/facts/Grade_(slope)/SlyxwANp">elevation grade</a> is a slope used to indicate the steepness of roads, paths and railway lines.</li>
<li>The <a href="/facts/Spread_(rational_trigonometry)/snPXwouV">spread</a> between two lines is defined in <a href="/facts/Rational_geometry/snPXwouV">rational geometry</a> as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.</li>
<li>Although done rarely, one can report the direct results of <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a>, such as the <a href="/facts/Sine/gQ6LXK8z">sine</a> of the angle.</li></ul>
<h2 id="angles-between-curves">Angles between curves</h2>

The angle between a line and a <a href="/facts/Curve/xBQ0pTBW">curve</a> (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the <a href="/facts/Tangent/sF0jH3EI">tangents</a> at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a>

<h2 id="bisecting-and-trisecting-angles">Bisecting and trisecting angles</h2>
Main articles: <a href="/facts/Bisection/JTsya7Ca">Bisection § Angle bisector</a>, and <a href="/facts/Angle_trisection/zSf98g6b">Angle trisection</a>
The <a href="/facts/Greek_mathematics/ealke3z8">ancient Greek mathematicians</a> knew how to bisect an angle (divide it into two angles of equal measure) using only a <a href="/facts/Compass_and_straightedge/lbaHf2m7">compass and straightedge</a> but could only trisect certain angles. In 1837, <a href="/facts/Pierre_Wantzel/gKVItq1G">Pierre Wantzel</a> showed that this construction could not be performed for most angles.

<h2 id="dot-product-and-generalisations">Dot product and generalisations </h2>
In the <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, the angle θ between two <a href="/facts/Euclidean_vector/bCLrdksy">Euclidean vectors</a> u and v is related to their <a href="/facts/Dot_product/tNz8MLjT">dot product</a> and their lengths by the formula

u
        
        ⋅
        
          v
        
        =
        cos
        ⁡
        (
        θ
        )
        
          ‖
          
            u
          
          ‖
        
        
          ‖
          
            v
          
          ‖
        
        .
      
    
    {\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}

This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their <a href="/facts/Normal_vector/jhtLIhcu">normal vectors</a> and between <a href="/facts/Skew_lines/2GSpRI8k">skew lines</a> from their vector equations.

<h3>Inner product</h3>
To define angles in an abstract real <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a>, we replace the Euclidean dot product ( · ) by the inner product 
 
 
 
 ⟨
 ⋅
 ,
 ⋅
 ⟩
 
 
 {\displaystyle \langle \cdot ,\cdot \rangle }
 
, i.e.

⟨
        
          u
        
        ,
        
          v
        
        ⟩
        =
        cos
        ⁡
        (
        θ
        )
         
        
          ‖
          
            u
          
          ‖
        
        
          ‖
          
            v
          
          ‖
        
        .
      
    
    {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}

In a complex <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a>, the expression for the cosine above may give non-real values, so it is replaced with

Re
        ⁡
        
          (
          
            ⟨
            
              u
            
            ,
            
              v
            
            ⟩
          
          )
        
        =
        cos
        ⁡
        (
        θ
        )
        
          ‖
          
            u
          
          ‖
        
        
          ‖
          
            v
          
          ‖
        
        .
      
    
    {\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}

or, more commonly, using the absolute value, with

|
          
            ⟨
            
              u
            
            ,
            
              v
            
            ⟩
          
          |
        
        =
        
          |
          
            cos
            ⁡
            (
            θ
            )
          
          |
        
        
          ‖
          
            u
          
          ‖
        
        
          ‖
          
            v
          
          ‖
        
        .
      
    
    {\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces 
 
 
 
 span
 ⁡
 (
 
 u
 
 )
 
 
 {\displaystyle \operatorname {span} (\mathbf {u} )}
 
 and 
 
 
 
 span
 ⁡
 (
 
 v
 
 )
 
 
 {\displaystyle \operatorname {span} (\mathbf {v} )}
 
 spanned by the vectors 
 
 
 
 
 u
 
 
 
 {\displaystyle \mathbf {u} }
 
 and 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 correspondingly.

<h3>Angles between subspaces</h3>
The definition of the angle between one-dimensional subspaces 
 
 
 
 span
 ⁡
 (
 
 u
 
 )
 
 
 {\displaystyle \operatorname {span} (\mathbf {u} )}
 
 and 
 
 
 
 span
 ⁡
 (
 
 v
 
 )
 
 
 {\displaystyle \operatorname {span} (\mathbf {v} )}
 
 given by

|
          
            ⟨
            
              u
            
            ,
            
              v
            
            ⟩
          
          |
        
        =
        
          |
          
            cos
            ⁡
            (
            θ
            )
          
          |
        
        
          ‖
          
            u
          
          ‖
        
        
          ‖
          
            v
          
          ‖
        
      
    
    {\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}

in a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> can be extended to subspaces of finite number of dimensions. Given two subspaces 
 
 
 
 
 
 U
 
 
 
 
 {\displaystyle {\mathcal {U}}}
 
, 
 
 
 
 
 
 W
 
 
 
 
 {\displaystyle {\mathcal {W}}}
 
 with 
 
 
 
 dim
 ⁡
 (
 
 
 U
 
 
 )
 :=
 k
 ≤
 dim
 ⁡
 (
 
 
 W
 
 
 )
 :=
 l
 
 
 {\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}
 
, this leads to a definition of 
 
 
 
 k
 
 
 {\displaystyle k}
 
 angles called canonical or <a href="/facts/Principal_angles/JMrfQQfL">principal angles</a> between subspaces.

<h3>Angles in Riemannian geometry</h3>
In <a href="/facts/Riemannian_geometry/pPEzjyQC">Riemannian geometry</a>, the <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> is used to define the angle between two <a href="/facts/Tangent/sF0jH3EI">tangents</a>. Where U and V are tangent vectors and gij are the components of the metric tensor G,

cos
        ⁡
        θ
        =
        
          
            
              
                g
                
                  i
                  j
                
              
              
                U
                
                  i
                
              
              
                V
                
                  j
                
              
            
            
              
                |
                
                  
                    g
                    
                      i
                      j
                    
                  
                  
                    U
                    
                      i
                    
                  
                  
                    U
                    
                      j
                    
                  
                
                |
              
              
                |
                
                  
                    g
                    
                      i
                      j
                    
                  
                  
                    V
                    
                      i
                    
                  
                  
                    V
                    
                      j
                    
                  
                
                |
              
            
          
        
        .
      
    
    {\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.}

<h3>Hyperbolic angle</h3>
A <a href="/facts/Hyperbolic_angle/m4IaIh0g">hyperbolic angle</a> is an <a href="/facts/Argument_of_a_function/QkuPc28I">argument</a> of a <a href="/facts/Hyperbolic_function/Y4Cb5S35">hyperbolic function</a> just as the circular angle is the argument of a <a href="/facts/Circular_function/czej7ur0">circular function</a>. The comparison can be visualized as the size of the openings of a <a href="/facts/Hyperbolic_sector/PCEFy8fd">hyperbolic sector</a> and a <a href="/facts/Circular_sector/TqwCaEA5">circular sector</a> since the <a href="/facts/Area/KsNG1G9t">areas</a> of these sectors correspond to the angle magnitudes in each case.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a> Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as <a href="/facts/Infinite_series/yDnlsgIe">infinite series</a> in their angle argument, the circular ones are just <a href="/facts/Alternating_series/wXpYnQ8b">alternating series</a> forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> in <a href="/facts/Introduction_to_the_Analysis_of_the_Infinite/xUf64mey">Introduction to the Analysis of the Infinite</a> (1748).

<h2 id="history-and-etymology">History and etymology</h2>
The word angle comes from the <a href="/facts/Latin/pM3Kge85">Latin</a> word angulus, meaning "corner". <a href="/facts/Cognate/nXETf5Y4">Cognate</a> words include the <a href="/facts/Greek_language/pzDd6eIz">Greek</a> ἀγκύλος (ankylοs) meaning "crooked, curved" and the <a href="/facts/English_language/tpkxGt7b">English</a> word "<a href="/facts/Ankle/jk7FpfJs">ankle</a>". Both are connected with the <a href="/facts/Proto-Indo-European_language/cVLNoM3p">Proto-Indo-European</a> root *ank-, meaning "to bend" or "bow".<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a>
<a href="/facts/Euclid/dHpVyL4R">Euclid</a> defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician <a href="/facts/Proclus/DRh6TKdX">Proclus</a>, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by <a href="/facts/Eudemus_of_Rhodes/M5QkuCgq">Eudemus of Rhodes</a>, who regarded an angle as a deviation from a <a href="/facts/Straight_line/gJqsr4Gj">straight line</a>; the second, angle as quantity, by <a href="/facts/Carpus_of_Antioch/sHAyjj5g">Carpus of Antioch</a>, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a>

<h3>Vertical angle theorem</h3>
The equality of vertically opposite angles is called the vertical angle theorem. <a href="/facts/Eudemus_of_Rhodes/M5QkuCgq">Eudemus of Rhodes</a> attributed the proof to <a href="/facts/Thales/zLoJUcTa">Thales of Miletus</a>.<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a><a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a> when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:

<ul><li>All straight angles are equal.</li>
<li>Equals added to equals are equal.</li>
<li>Equals subtracted from equals are equal.</li></ul>
When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 180° − x. Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.

<h2 id="angles-in-geography-and-astronomy">Angles in geography and astronomy</h2>
In <a href="/facts/Geography/TRd1Toyo">geography</a>, the location of any point on the Earth can be identified using a <a href="/facts/Geographic_coordinate_system/RkjfcY6k">geographic coordinate system</a>. This system specifies the <a href="/facts/Latitude/IBYqkjhU">latitude</a> and <a href="/facts/Longitude/HsC61ML9">longitude</a> of any location in terms of angles subtended at the center of the Earth, using the <a href="/facts/Equator/CjaxbVaN">equator</a> and (usually) the <a href="/facts/Greenwich_meridian/5dhcewNL">Greenwich meridian</a> as references.
In <a href="/facts/Astronomy/0ArJVM1p">astronomy</a>, a given point on the <a href="/facts/Celestial_sphere/TYAtXgor">celestial sphere</a> (that is, the apparent position of an astronomical object) can be identified using any of several <a href="/facts/Astronomical_coordinate_systems/23HhqvHj">astronomical coordinate systems</a>, where the references vary according to the particular system. Astronomers measure the <a href="/facts/Angular_separation/6ob2n0Wb">angular separation</a> of two <a href="/facts/Star/euBFuo07">stars</a> by imagining two lines through the center of the <a href="/facts/Earth/HLLmDfj0">Earth</a>, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.
In both geography and astronomy, a sighting direction can be specified in terms of a <a href="/facts/Vertical_angle/gFUTEQYq">vertical angle</a> such as <a href="/facts/Altitude_angle/msJQPYg0">altitude</a> /<a href="/facts/Elevation_angle/5cVnNof3">elevation</a> with respect to the <a href="/facts/Horizon/LxwxWfC2">horizon</a> as well as the <a href="/facts/Azimuth/kEyt9nJ3">azimuth</a> with respect to <a href="/facts/North/Hty6w98C">north</a>.
Astronomers also measure objects' apparent size as an <a href="/facts/Angular_diameter/p2xv4vcQ">angular diameter</a>. For example, the <a href="/facts/Full_moon/NRMWFPh2">full moon</a> has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The <a href="/facts/Small-angle_formula/8PQygrBq">small-angle formula</a> can convert such an angular measurement into a distance/size ratio.
Other astronomical approximations include:

<ul><li>0.5° is the approximate diameter of the <a href="/facts/Sun/CadPCVHh">Sun</a> and of the <a href="/facts/Moon/zKHqEMPb">Moon</a> as viewed from Earth.</li>
<li>1° is the approximate width of the <a href="/facts/Little_finger/094Gy4Ra">little finger</a> at arm's length.</li>
<li>10° is the approximate width of a closed fist at arm's length.</li>
<li>20° is the approximate width of a handspan at arm's length.</li></ul>
These measurements depend on the individual subject, and the above should be treated as rough <a href="/facts/Rule_of_thumb/RopfGsJ0">rule of thumb</a> approximations only.
In astronomy, <a href="/facts/Right_ascension/z4CEsTFu">right ascension</a> and <a href="/facts/Declination/5X4S4k9u">declination</a> are usually measured in angular units, expressed in terms of time, based on a 24-hour day.

<table><tbody><tr><th>Unit</th><th><a href="/facts/Sexagesimal/bspNRWdR">Symbol</a></th><th>Degrees</th><th>Radians</th><th>Turns</th><th>Other</th></tr><tr><th>Hour</th><td>h</td><td>15°</td><td>π⁄12 rad</td><td>1⁄24 turn</td><td></td></tr><tr><th>Minute</th><td>m</td><td>0°15′</td><td>π⁄720 rad</td><td>1⁄1440 turn</td><td>1⁄60 hour</td></tr><tr><th>Second</th><td>s</td><td>0°0′15″</td><td>π⁄43200 rad</td><td>1⁄86400 turn</td><td>1⁄60 minute</td></tr></tbody></table>
<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Angle_measuring_instrument/DoKGJII2">Angle measuring instrument</a></li>
<li><a href="/facts/Angles_between_flats/JMrfQQfL">Angles between flats</a></li>
<li><a href="/facts/Angular_statistics/ucnUv52T">Angular statistics</a> (<a href="/facts/Angular_mean/eMWmRu8P">mean</a>, <a href="/facts/Angular_standard_deviation/ucnUv52T">standard deviation</a>)</li>
<li><a href="/facts/Bisection/JTsya7Ca">Angle bisector</a></li>
<li><a href="/facts/Angular_acceleration/jDv9uzAS">Angular acceleration</a></li>
<li><a href="/facts/Angular_diameter/p2xv4vcQ">Angular diameter</a></li>
<li><a href="/facts/Angular_velocity/2n8JljAT">Angular velocity</a></li>
<li><a href="/facts/Argument_(complex_analysis)/JSjfxb5U">Argument (complex analysis)</a></li>
<li><a href="/facts/Astrological_aspect/lxVn5ApM">Astrological aspect</a></li>
<li><a href="/facts/Central_angle/h00sYLNj">Central angle</a></li>
<li><a href="/facts/Clock_angle_problem/92EsUzVA">Clock angle problem</a></li>
<li><a href="/facts/Decimal_degrees/kNoEsSDG">Decimal degrees</a></li>
<li><a href="/facts/Dihedral_angle/C9WEJsuS">Dihedral angle</a></li>
<li><a href="/facts/Exterior_angle_theorem/a2qH6823">Exterior angle theorem</a></li>
<li><a href="/facts/Golden_angle/5HGPvRQC">Golden angle</a></li>
<li><a href="/facts/Great_circle_distance/s6HNWn8e">Great circle distance</a></li>
<li><a href="/facts/Horn_angle/sEWPvc0K">Horn angle</a></li>
<li><a href="/facts/Inscribed_angle/fTYeVKxt">Inscribed angle</a></li>
<li><a href="/facts/Irrational_angle/1xQtFceA">Irrational angle</a></li>
<li><a href="/facts/Phase_(waves)/cpGtuVll">Phase (waves)</a></li>
<li><a href="/facts/Protractor/6INTgy5h">Protractor</a></li>
<li><a href="/facts/Solid_angle/se8IXufg">Solid angle</a></li>
<li><a href="/facts/Spherical_angle/ekVT5dW2">Spherical angle</a></li>
<li><a href="/facts/Subtended_angle/GQxln7lP">Subtended angle</a></li>
<li><a href="/facts/Tangential_angle/HGw25Uum">Tangential angle</a></li>
<li><a href="/facts/Transcendent_angle/bA0B2KcC">Transcendent angle</a></li>
<li><a href="/facts/Trisection/zSf98g6b">Trisection</a></li>
<li><a href="/facts/Zenith_angle/tcm21e6t">Zenith angle</a></li></ul>

<h2 id="notes">Notes</h2>

<h2 id="bibliography">Bibliography</h2>

<ul><li>Aboughantous, Charles H. (2010), <a href="https://books.google.com/books?id=4JK19X5ZI2IC&pg=PA18">A High School First Course in Euclidean Plane Geometry</a>, Universal Publishers, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-59942-822-2</li>
<li>Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly". American Journal of Physics. 4 (4): 175–179. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1936AmJPh...4..175B">1936AmJPh...4..175B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1119%2F1.1999110">10.1119/1.1999110</a>.</li>
<li>Brownstein, K. R. (July 1997). <a href="https://doi.org/10.1119%2F1.18616">"Angles—Let's treat them squarely"</a>. American Journal of Physics. 65 (7): 605–614. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1997AmJPh..65..605B">1997AmJPh..65..605B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1119%2F1.18616">10.1119/1.18616</a>.</li>
<li>Eder, W E (January 1982). "A Viewpoint on the Quantity "Plane Angle"". Metrologia. 18 (1): 1–12. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1982Metro..18....1E">1982Metro..18....1E</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0026-1394%2F18%2F1%2F002">10.1088/0026-1394/18/1/002</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:250750831">250750831</a>.</li>
<li>Foster, Marcus P (1 December 2010). <a href="https://publications.csiro.au/rpr/download?pid=csiro:EP11794&dsid=DS4">"The next 50 years of the SI: a review of the opportunities for the e-Science age"</a>. Metrologia. 47 (6): R41 – R51. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0026-1394%2F47%2F6%2FR01">10.1088/0026-1394/47/6/R01</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:117711734">117711734</a>.</li>
<li>Godfrey, Charles; Siddons, A. W. (1919), <a href="https://archive.org/details/elementarygeomet00godfuoft/page/9/mode/2up?">Elementary geometry: practical and theoretical</a> (3rd ed.), Cambridge University Press</li>
<li>Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry / Euclidean and Non-Euclidean with History (3rd ed.), Pearson Prentice Hall, p. 104, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-143748-7</li>
<li>Heiberg, Johan Ludvig (1908), <a href="/facts/T._L._Heath/E9xE228u">Heath, T. L.</a> (ed.), <a href="https://books.google.com/books?id=UhgPAAAAIAAJ">Euclid</a>, The Thirteen Books of Euclid's Elements, vol. 1, <a href="/facts/Cambridge%2c_England/gNRdbmwK">Cambridge</a>: Cambridge University Press.</li>
<li>Jacobs, Harold R. (1974), Geometry, W. H. Freeman, pp. 97, 255, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7167-0456-0</li>
<li>Leonard, B P (1 October 2021). "Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)". Metrologia. 58 (5): 052001. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2021Metro..58e2001L">2021Metro..58e2001L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F1681-7575%2Fabe0fc">10.1088/1681-7575/abe0fc</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:234036217">234036217</a>.</li>
<li>Lévy-Leblond, Jean-Marc (September 1998). <a href="https://www.researchgate.net/publication/253371022">"Dimensional angles and universal constants"</a>. American Journal of Physics. 66 (9): 814–815. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1998AmJPh..66..814L">1998AmJPh..66..814L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1119%2F1.18964">10.1119/1.18964</a>.</li>
<li>Mills, Ian (1 June 2016). "On the units radian and cycle for the quantity plane angle". Metrologia. 53 (3): 991–997. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2016Metro..53..991M">2016Metro..53..991M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0026-1394%2F53%2F3%2F991">10.1088/0026-1394/53/3/991</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:126032642">126032642</a>.</li>
<li>Mohr, Peter J; Phillips, William D (1 February 2015). <a href="https://doi.org/10.1088%2F0026-1394%2F52%2F1%2F40">"Dimensionless units in the SI"</a>. Metrologia. 52 (1): 40–47. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1409.2794">1409.2794</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2015Metro..52...40M">2015Metro..52...40M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0026-1394%2F52%2F1%2F40">10.1088/0026-1394/52/1/40</a>.</li>
<li>Mohr, Peter J; Shirley, Eric L; Phillips, William D; Trott, Michael (23 June 2022). <a href="https://doi.org/10.1088%2F1681-7575%2Fac7bc2">"On the dimension of angles and their units"</a>. Metrologia. 59 (5): 053001. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2203.12392">2203.12392</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2022Metro..59e3001M">2022Metro..59e3001M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F1681-7575%2Fac7bc2">10.1088/1681-7575/ac7bc2</a>.</li>
<li>Moser, James M. (1971), <a href="https://archive.org/details/modernelementary0000mose/page/41/mode/2up">Modern Elementary Geometry</a>, Prentice-Hall</li>
<li>Quincey, Paul (1 April 2016). "The range of options for handling plane angle and solid angle within a system of units". Metrologia. 53 (2): 840–845. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2016Metro..53..840Q">2016Metro..53..840Q</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0026-1394%2F53%2F2%2F840">10.1088/0026-1394/53/2/840</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:125438811">125438811</a>.</li>
<li>Quincey, Paul (1 October 2021). "Angles in the SI: a detailed proposal for solving the problem". Metrologia. 58 (5): 053002. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2108.05704">2108.05704</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2021Metro..58e3002Q">2021Metro..58e3002Q</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F1681-7575%2Fac023f">10.1088/1681-7575/ac023f</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:236547235">236547235</a>.</li>
<li>Romain, Jacques E. (July 1962). <a href="https://doi.org/10.6028%2Fjres.066B.012">"Angle as a fourth fundamental quantity"</a>. Journal of Research of the National Bureau of Standards Section B. 66B (3): 97. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.6028%2Fjres.066B.012">10.6028/jres.066B.012</a>.</li>
<li>Sidorov, L. A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Angle">"Angle"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Slocum, Jonathan (2007), <a href="https://web.archive.org/web/20100627012240/http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html">Preliminary Indo-European lexicon — Pokorny PIE data</a>, <a href="/facts/Linguistics_Research_Center_at_UT_Austin/hUoPdd7d">University of Texas research department: linguistics research center</a>, archived from <a href="http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html">the original</a> on 27 June 2010, retrieved 2 Feb 2010</li>
<li>Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27</li>
<li>Torrens, A B (1 January 1986). "On Angles and Angular Quantities". Metrologia. 22 (1): 1–7. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1986Metro..22....1T">1986Metro..22....1T</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0026-1394%2F22%2F1%2F002">10.1088/0026-1394/22/1/002</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:250801509">250801509</a>.</li>
<li>Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines", New Century Mathematics, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-800177-5</li></ul>
 This article incorporates text from a publication now in the <a href="/facts/Public_domain/YWKMDKw4">public domain</a>: <a href="/facts/Hugh_Chisholm/2vWblUBe">Chisholm, Hugh</a>, ed. (1911), "Angle", <a href="/facts/Encyclop%25C3%25A6dia_Britannica_Eleventh_Edition/mPElbCRY">Encyclopædia Britannica</a>, vol. 2 (11th ed.), Cambridge University Press, p. 14

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Angles (geometry).

The Wikibook Geometry has a page on the topic of: Unified Angles
 
<ul><li><a href="https://en.wikisource.org/wiki/Encyclop%C3%A6dia_Britannica,_Ninth_Edition/Angle">"Angle" </a>, <a href="/facts/Encyclop%25C3%25A6dia_Britannica/CEyVsjlN">Encyclopædia Britannica</a>, vol. 2 (9th ed.), 1878, pp. 29–30</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hilbert, David. The Foundations of Geometry (PDF). p. 9. <a href="https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf" target="_blank">https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Sidorov 2001 - Sidorov, L. A. (2001) [1994], "Angle", Encyclopedia of Mathematics, EMS Press <a href="https://www.encyclopediaofmath.org/index.php?title=Angle" target="_blank">https://www.encyclopediaofmath.org/index.php?title=Angle</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Evgrafov, M. A. (2019-09-18). Analytic Functions. Courier Dover Publications. ISBN 978-0-486-84366-7. <a href="978-0-486-84366-7" target="_blank">978-0-486-84366-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Papadopoulos, Athanase (2012). Strasbourg Master Class on Geometry. European Mathematical Society. ISBN 978-3-03719-105-7. <a href="978-3-03719-105-7" target="_blank">978-3-03719-105-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">An angular sector can be constructed by the combination of two rotated half-planes, either their intersection or union (in the case of acute or obtuse angles, respectively).[5][6] It corresponds to a circular sector of infinite radius and a flat pencil of half-lines.[7] <a href="/wiki/Half-plane" target="_blank">/wiki/Half-plane</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Hilbert, David. The Foundations of Geometry (PDF). p. 9. <a href="https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf" target="_blank">https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Aboughantous 2010, p. 18. - Aboughantous, Charles H. (2010), A High School First Course in Euclidean Plane Geometry, Universal Publishers, ISBN 978-1-59942-822-2 <a href="https://books.google.com/books?id=4JK19X5ZI2IC&pg=PA18" target="_blank">https://books.google.com/books?id=4JK19X5ZI2IC&pg=PA18</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Moser 1971, p. 41. - Moser, James M. (1971), Modern Elementary Geometry, Prentice-Hall <a href="https://archive.org/details/modernelementary0000mose/page/41/mode/2up" target="_blank">https://archive.org/details/modernelementary0000mose/page/41/mode/2up</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Godfrey & Siddons 1919, p. 9. - Godfrey, Charles; Siddons, A. W. (1919), Elementary geometry: practical and theoretical (3rd ed.), Cambridge University Press <a href="https://archive.org/details/elementarygeomet00godfuoft/page/9/mode/2up?" target="_blank">https://archive.org/details/elementarygeomet00godfuoft/page/9/mode/2up?</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Moser 1971, p. 71. - Moser, James M. (1971), Modern Elementary Geometry, Prentice-Hall <a href="https://archive.org/details/modernelementary0000mose/page/41/mode/2up" target="_blank">https://archive.org/details/modernelementary0000mose/page/41/mode/2up</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Godfrey & Siddons 1919, p. 9. - Godfrey, Charles; Siddons, A. W. (1919), Elementary geometry: practical and theoretical (3rd ed.), Cambridge University Press <a href="https://archive.org/details/elementarygeomet00godfuoft/page/9/mode/2up?" target="_blank">https://archive.org/details/elementarygeomet00godfuoft/page/9/mode/2up?</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Moser 1971, p. 41. - Moser, James M. (1971), Modern Elementary Geometry, Prentice-Hall <a href="https://archive.org/details/modernelementary0000mose/page/41/mode/2up" target="_blank">https://archive.org/details/modernelementary0000mose/page/41/mode/2up</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Moise, Edwin, E (1990). Elementary geometry from an advanced standpoint (PDF) (3rd ed.). Addison-Wesley Publishing Company. p. 96.{{cite book}}: CS1 maint: multiple names: authors list (link) <a href="https://www.ime.usp.br/~toscano/disc/2021/Moise.pdf" target="_blank">https://www.ime.usp.br/~toscano/disc/2021/Moise.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Wong & Wong 2009, pp. 161–163 - Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines", New Century Mathematics, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, ISBN 978-0-19-800177-5 <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Jacobs 1974, p. 255. - Jacobs, Harold R. (1974), Geometry, W. H. Freeman, pp. 97, 255, ISBN 978-0-7167-0456-0 <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">"Complementary Angles". www.mathsisfun.com. Retrieved 2020-08-17. <a href="https://www.mathsisfun.com/geometry/complementary-angles.html" target="_blank">https://www.mathsisfun.com/geometry/complementary-angles.html</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Chisholm 1911 - Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, vol. 2 (11th ed.), Cambridge University Press, p. 14 <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">"Supplementary Angles". www.mathsisfun.com. Retrieved 2020-08-17. <a href="https://www.mathsisfun.com/geometry/supplementary-angles.html" target="_blank">https://www.mathsisfun.com/geometry/supplementary-angles.html</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Jacobs 1974, p. 97. - Jacobs, Harold R. (1974), Geometry, W. H. Freeman, pp. 97, 255, ISBN 978-0-7167-0456-0 <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Willis, Clarence Addison (1922). Plane Geometry. Blakiston's Son. p. 8. <a href="https://archive.org/details/planegeometryexp00willrich/page/8/" target="_blank">https://archive.org/details/planegeometryexp00willrich/page/8/</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Henderson & Taimina 2005, p. 104. - Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry / Euclidean and Non-Euclidean with History (3rd ed.), Pearson Prentice Hall, p. 104, ISBN 978-0-13-143748-7 <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007. <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007. <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007. <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">D. Zwillinger, ed. (1995), CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, p. 270 as cited in Weisstein, Eric W. "Exterior Angle". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Chisholm 1911 - Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, vol. 2 (11th ed.), Cambridge University Press, p. 14 <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">International Bureau of Weights and Measures (20 May 2019), The International System of Units (SI) (PDF) (9th ed.), ISBN 978-92-822-2272-0, archived from the original on 18 October 2021 <a href="978-92-822-2272-0" target="_blank">978-92-822-2272-0</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">This approach requires, however, an additional proof that the measure of the angle does not change with changing radius r, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.[24] <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">"angular unit". TheFreeDictionary.com. Retrieved 2020-08-31. <a href="https://www.thefreedictionary.com/angular+unit" target="_blank">https://www.thefreedictionary.com/angular+unit</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">"ooPIC Programmer's Guide - Chapter 15: URCP". ooPIC Manual & Technical Specifications - ooPIC Compiler Ver 6.0. Savage Innovations, LLC. 2007 [1997]. Archived from the original on 2008-06-28. Retrieved 2019-08-05. <a href="https://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm" target="_blank">https://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Jeans, James Hopwood (1947). The Growth of Physical Science. CUP Archive. p. 7. <a href="/wiki/James_Hopwood_Jeans" target="_blank">/wiki/James_Hopwood_Jeans</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2. <a href="/wiki/Francis_Dominic_Murnaghan_(mathematician)" target="_blank">/wiki/Francis_Dominic_Murnaghan_(mathematician)</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">"Mathwords: Reference Angle". www.mathwords.com. Archived from the original on 23 October 2017. Retrieved 26 April 2018. <a href="http://www.mathwords.com/r/reference_angle.htm" target="_blank">http://www.mathwords.com/r/reference_angle.htm</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">McKeague, Charles P. (2008). Trigonometry (6th ed.). Belmont, CA: Thomson Brooks/Cole. p. 110. ISBN 978-0495382607. <a href="978-0495382607" target="_blank">978-0495382607</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Chisholm 1911; Heiberg 1908, p. 178 - Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, vol. 2 (11th ed.), Cambridge University Press, p. 14 <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Robert Baldwin Hayward (1892) The Algebra of Coplanar Vectors and Trigonometry, chapter six <a href="/wiki/Robert_Baldwin_Hayward" target="_blank">/wiki/Robert_Baldwin_Hayward</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Slocum 2007 - Slocum, Jonathan (2007), Preliminary Indo-European lexicon — Pokorny PIE data, University of Texas research department: linguistics research center, archived from the original on 27 June 2010, retrieved 2 Feb 2010 <a href="https://web.archive.org/web/20100627012240/http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html" target="_blank">https://web.archive.org/web/20100627012240/http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Chisholm 1911; Heiberg 1908, pp. 177–178 - Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, vol. 2 (11th ed.), Cambridge University Press, p. 14 <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Euclid. The Elements. Proposition I:13. <a href="/wiki/Euclid" target="_blank">/wiki/Euclid</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Shute, Shirk & Porter 1960, pp. 25–27. - Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27 <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Shute, Shirk & Porter 1960, pp. 25–27. - Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27 <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
</ol>

Angle open-in-new

Angle