Cone

<h2 id="further-terminology">Further terminology </h2>
<p>The perimeter of the base of a cone is called the <i>directrix</i>, and each of the line segments between the directrix and apex is a <i>generatrix</i> or <i>generating line</i> of the lateral surface.  (For the connection between this sense of the term <i>directrix</i> and the <a href="/facts/Directrix_(conic_section)/teqgLptX">directrix</a> of a conic section, see <a href="/facts/Dandelin_spheres/fE2XYh1T">Dandelin spheres</a>.)
</p><p>The <i>base radius</i> of a circular cone is the <a href="/facts/Radius/9Gr9deYD">radius</a> of its base; often this is simply called the radius of the cone. The <i>aperture</i> of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle <i>θ</i> to the axis, the aperture is 2<i>θ</i>. In <a href="/facts/Optics/p0vq4S1T">optics</a>, the angle <i>θ</i> is called the <i>half-angle</i> of the cone, to distinguish it from the aperture.
</p>

<p>A cone with a region including its apex cut off by a plane is called a <i>truncated cone</i>; if the <a href="/facts/Truncation_(geometry)/DYb589Je">truncation</a> plane is parallel to the cone's base, it is called a <i><a href="/facts/Frustum/TEfLWynX">frustum</a></i>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>  An <i><a href="/facts/Elliptical_cone/2c9GIzYD">elliptical cone</a></i> is a cone with an <a href="/facts/Ellipse/1wUDnGxY">elliptical</a> base.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>  A <i>generalized cone</i> is the surface created by the set of lines passing through a vertex and every point on a boundary (see <a href="/facts/Visual_hull/yrqyp7L9">Visual hull</a>).
</p>
<h2 id="measurements-and-equations">Measurements and equations</h2>
<h3>Volume</h3>

<p>The <a href="/facts/Volume/aeAXCBUd">volume</a> 
  
    
      
        V
      
    
    {\displaystyle V}
  
 of any conic solid is one third of the product of the area of the base 
  
    
      
        
          A
          
            B
          
        
      
    
    {\displaystyle A_{B}}
  
 and the height 
  
    
      
        h
      
    
    {\displaystyle h}
  
<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 
</p><p>
  
    
      
        V
        =
        
          
            1
            3
          
        
        
          A
          
            B
          
        
        h
        .
      
    
    {\displaystyle V={\frac {1}{3}}A_{B}h.}

</p><p>In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral
</p><p>
  
    
      
        ∫
        
          x
          
            2
          
        
        
        d
        x
        =
        
          
            
              1
              3
            
          
        
        
          x
          
            3
          
        
      
    
    {\displaystyle \int x^{2}\,dx={\tfrac {1}{3}}x^{3}}

</p><p>Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying <a href="/facts/Cavalieri%2527s_principle/CRSyxWzD">Cavalieri's principle</a> – specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the <a href="/facts/Method_of_exhaustion/jyzkY44y">method of exhaustion</a>. This is essentially the content of <a href="/facts/Hilbert%2527s_third_problem/UcS6SWVR">Hilbert's third problem</a> – more precisely, not all polyhedral pyramids are <i>scissors congruent</i> (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<h3>Center of mass</h3>
<p>The <a href="/facts/Center_of_mass/n4Lr9GAz">center of mass</a> of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.
</p>
<h3>Right circular cone</h3>
<h4>Volume</h4>
<p>For a circular cone with radius 
  
    
      
        r
      
    
    {\displaystyle r}
  
 and height 
  
    
      
        h
      
    
    {\displaystyle h}
  
, the base is a circle of area 
  
    
      
        π
        
          r
          
            2
          
        
      
    
    {\displaystyle \pi r^{2}}
  
 thus the formula for volume is:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p><p>
  
    
      
        V
        =
        
          
            1
            3
          
        
        π
        
          r
          
            2
          
        
        h
      
    
    {\displaystyle V={\frac {1}{3}}\pi r^{2}h}

</p>
<h4>Slant height</h4>
<p>The <a href="/facts/Slant_height/2c9GIzYD">slant height</a> of a right circular cone is the distance from any point on the <a href="/facts/Circle/9fy5ggKn">circle</a> of its base to the apex via a line segment along the surface of the cone. It is given by 
  
    
      
        
          
            
              r
              
                2
              
            
            +
            
              h
              
                2
              
            
          
        
      
    
    {\displaystyle {\sqrt {r^{2}+h^{2}}}}
  
, where 
  
    
      
        r
      
    
    {\displaystyle r}
  
 is the <a href="/facts/Radius/9Gr9deYD">radius</a> of the base and 
  
    
      
        h
      
    
    {\displaystyle h}
  
 is the height. This can be proved by the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a>.
</p>
<h4>Surface area</h4>
<p>The <a href="/facts/Lateral_surface/edq0pyHv">lateral surface</a> area of a right circular cone is 
  
    
      
        L
        S
        A
        =
        π
        r
        ℓ
      
    
    {\displaystyle LSA=\pi r\ell }
  
 where 
  
    
      
        r
      
    
    {\displaystyle r}
  
 is the radius of the circle at the bottom of the cone and 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
 is the slant height of the cone.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>  The surface area of the bottom circle of a cone is the same as for any circle, 
  
    
      
        π
        
          r
          
            2
          
        
      
    
    {\displaystyle \pi r^{2}}
  
.  Thus, the total surface area of a right circular cone can be expressed as each of the following:
</p>
<ul><li>Radius and height</li></ul>

π
        
          r
          
            2
          
        
        +
        π
        r
        
          
            
              r
              
                2
              
            
            +
            
              h
              
                2
              
            
          
        
      
    
    {\displaystyle \pi r^{2}+\pi r{\sqrt {r^{2}+h^{2}}}}

(the area of the base plus the area of the lateral surface; the term 
  
    
      
        
          
            
              r
              
                2
              
            
            +
            
              h
              
                2
              
            
          
        
      
    
    {\displaystyle {\sqrt {r^{2}+h^{2}}}}
  
 is the slant height)

π
        r
        
          (
          
            r
            +
            
              
                
                  r
                  
                    2
                  
                
                +
                
                  h
                  
                    2
                  
                
              
            
          
          )
        
      
    
    {\displaystyle \pi r\left(r+{\sqrt {r^{2}+h^{2}}}\right)}

where 
  
    
      
        r
      
    
    {\displaystyle r}
  
 is the radius and 
  
    
      
        h
      
    
    {\displaystyle h}
  
 is the height.

<ul><li>Radius and slant height</li></ul>

π
        
          r
          
            2
          
        
        +
        π
        r
        ℓ
      
    
    {\displaystyle \pi r^{2}+\pi r\ell }

π
        r
        (
        r
        +
        ℓ
        )
      
    
    {\displaystyle \pi r(r+\ell )}

where 
  
    
      
        r
      
    
    {\displaystyle r}
  
 is the radius and 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
 is the slant height.
<ul><li>Circumference and slant height</li></ul>

c
              
                2
              
            
            
              4
              π
            
          
        
        +
        
          
            
              c
              ℓ
            
            2
          
        
      
    
    {\displaystyle {\frac {c^{2}}{4\pi }}+{\frac {c\ell }{2}}}

(
          
            
              c
              2
            
          
          )
        
        
          (
          
            
              
                c
                
                  2
                  π
                
              
            
            +
            ℓ
          
          )
        
      
    
    {\displaystyle \left({\frac {c}{2}}\right)\left({\frac {c}{2\pi }}+\ell \right)}

where 
  
    
      
        c
      
    
    {\displaystyle c}
  
 is the circumference and 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
 is the slant height.
<ul><li>Apex angle and height</li></ul>

π
        
          h
          
            2
          
        
        tan
        ⁡
        
          
            θ
            2
          
        
        
          (
          
            tan
            ⁡
            
              
                θ
                2
              
            
            +
            sec
            ⁡
            
              
                θ
                2
              
            
          
          )
        
      
    
    {\displaystyle \pi h^{2}\tan {\frac {\theta }{2}}\left(\tan {\frac {\theta }{2}}+\sec {\frac {\theta }{2}}\right)}

−
        
          
            
              π
              
                h
                
                  2
                
              
              sin
              ⁡
              
                
                  θ
                  2
                
              
            
            
              sin
              ⁡
              
                
                  θ
                  2
                
              
              −
              1
            
          
        
      
    
    {\displaystyle -{\frac {\pi h^{2}\sin {\frac {\theta }{2}}}{\sin {\frac {\theta }{2}}-1}}}

where 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
 is the apex angle and 
  
    
      
        h
      
    
    {\displaystyle h}
  
 is the height.
<h4>Circular sector</h4>
<p>The <a href="/facts/Circular_sector/TqwCaEA5">circular sector</a> is obtained by unfolding the surface of one nappe of the cone:
</p>
<ul><li>radius <i>R</i></li></ul>

R
        =
        
          
            
              r
              
                2
              
            
            +
            
              h
              
                2
              
            
          
        
      
    
    {\displaystyle R={\sqrt {r^{2}+h^{2}}}}

<ul><li>arc length <i>L</i></li></ul>

L
        =
        c
        =
        2
        π
        r
      
    
    {\displaystyle L=c=2\pi r}

<ul><li>central angle <i>φ</i> in radians</li></ul>

φ
        =
        
          
            L
            R
          
        
        =
        
          
            
              2
              π
              r
            
            
              
                r
                
                  2
                
              
              +
              
                h
                
                  2
                
              
            
          
        
      
    
    {\displaystyle \varphi ={\frac {L}{R}}={\frac {2\pi r}{\sqrt {r^{2}+h^{2}}}}}

<h4>Equation form</h4>
<p>The surface of a cone can be parameterized as
</p>

f
        (
        θ
        ,
        h
        )
        =
        (
        h
        cos
        ⁡
        θ
        ,
        h
        sin
        ⁡
        θ
        ,
        h
        )
        ,
      
    
    {\displaystyle f(\theta ,h)=(h\cos \theta ,h\sin \theta ,h),}

<p>where 
  
    
      
        θ
        ∈
        [
        0
        ,
        2
        π
        )
      
    
    {\displaystyle \theta \in [0,2\pi )}
  
 is the angle "around" the cone, and 
  
    
      
        h
        ∈
        
          R
        
      
    
    {\displaystyle h\in \mathbb {R} }
  
 is the "height" along the cone.
</p><p>A right solid circular cone with height 
  
    
      
        h
      
    
    {\displaystyle h}
  
 and aperture  
  
    
      
        2
        θ
      
    
    {\displaystyle 2\theta }
  
, whose axis is the 
  
    
      
        z
      
    
    {\displaystyle z}
  
 coordinate axis and whose apex is the origin, is described parametrically as
</p>

F
        (
        s
        ,
        t
        ,
        u
        )
        =
        
          (
          
            u
            tan
            ⁡
            s
            cos
            ⁡
            t
            ,
            u
            tan
            ⁡
            s
            sin
            ⁡
            t
            ,
            u
          
          )
        
      
    
    {\displaystyle F(s,t,u)=\left(u\tan s\cos t,u\tan s\sin t,u\right)}

<p>where 
  
    
      
        s
        ,
        t
        ,
        u
      
    
    {\displaystyle s,t,u}
  
 range over 
  
    
      
        [
        0
        ,
        θ
        )
      
    
    {\displaystyle [0,\theta )}
  
, 
  
    
      
        [
        0
        ,
        2
        π
        )
      
    
    {\displaystyle [0,2\pi )}
  
, and 
  
    
      
        [
        0
        ,
        h
        ]
      
    
    {\displaystyle [0,h]}
  
, respectively.
</p><p>In <a href="/facts/Implicit_function/UJmQ2Cyk">implicit</a> form, the same solid is defined by the inequalities
</p>

{
        F
        (
        x
        ,
        y
        ,
        z
        )
        ≤
        0
        ,
        z
        ≥
        0
        ,
        z
        ≤
        h
        }
        ,
      
    
    {\displaystyle \{F(x,y,z)\leq 0,z\geq 0,z\leq h\},}

<p>where
</p>

F
        (
        x
        ,
        y
        ,
        z
        )
        =
        (
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        )
        (
        cos
        ⁡
        θ
        
          )
          
            2
          
        
        −
        
          z
          
            2
          
        
        (
        sin
        ⁡
        θ
        
          )
          
            2
          
        
        .
        
      
    
    {\displaystyle F(x,y,z)=(x^{2}+y^{2})(\cos \theta )^{2}-z^{2}(\sin \theta )^{2}.\,}

<p>More generally, a right circular cone with vertex at the origin, axis parallel to the vector 
  
    
      
        d
      
    
    {\displaystyle d}
  
, and aperture 
  
    
      
        2
        θ
      
    
    {\displaystyle 2\theta }
  
, is given by the implicit <a href="/facts/Vector_calculus/mXKIJLJE">vector</a> equation 
  
    
      
        F
        (
        u
        )
        =
        0
      
    
    {\displaystyle F(u)=0}
  
 where
</p>

F
        (
        u
        )
        =
        (
        u
        ⋅
        d
        
          )
          
            2
          
        
        −
        (
        d
        ⋅
        d
        )
        (
        u
        ⋅
        u
        )
        (
        cos
        ⁡
        θ
        
          )
          
            2
          
        
      
    
    {\displaystyle F(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \theta )^{2}}

F
        (
        u
        )
        =
        u
        ⋅
        d
        −
        
          |
        
        d
        
          |
        
        
          |
        
        u
        
          |
        
        cos
        ⁡
        θ
      
    
    {\displaystyle F(u)=u\cdot d-|d||u|\cos \theta }

<p>where 
  
    
      
        u
        =
        (
        x
        ,
        y
        ,
        z
        )
      
    
    {\displaystyle u=(x,y,z)}
  
, and 
  
    
      
        u
        ⋅
        d
      
    
    {\displaystyle u\cdot d}
  
 denotes the <a href="/facts/Dot_product/tNz8MLjT">dot product</a>.
</p>
<h2 id="projective-geometry">Projective geometry</h2>

<p>In <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a>, a <a href="/facts/Cylinder_(geometry)/0wQPPr3k">cylinder</a> is simply a cone whose apex is at infinity.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as <a href="/facts/Arctan/bVwU9FPG">arctan</a>, in the limit forming a <a href="/facts/Right_angle/psaslRq7">right angle</a>. This is useful in the definition of <a href="/facts/Degenerate_conic/9Ek4ja1g">degenerate conics</a>, which require considering the cylindrical conics.
</p><p>According to <a href="/facts/G._B._Halsted/40yRg39F">G. B. Halsted</a>, a cone is generated similarly to a <a href="/facts/Steiner_conic/hHMKh7UD">Steiner conic</a> only with a projectivity and <a href="/facts/Pencil_(mathematics)/b75bj3wd">axial pencils</a> (not in perspective) rather than the projective ranges used for the Steiner conic:
</p><p>"If two copunctual non-costraight axial pencils are projective but not perspective, the meets of correlated planes form a 'conic surface of the second order', or 'cone'."<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>
<h2 id="generalizations">Generalizations</h2>
<p class="note">Further information: <a href="/facts/Hypercone/YHJS9jzq">Hypercone</a></p>
<p>The definition of a cone may be extended to higher dimensions; see <a href="/facts/Convex_cone/h1tARfw4">convex cone</a>. In this case, one says that a <a href="/facts/Convex_set/vdAuJRJl">convex set</a> <i>C</i> in the <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 is a cone (with apex at the origin) if for every vector <i>x</i> in <i>C</i> and every nonnegative real number <i>a</i>, the vector <i>ax</i> is in <i>C</i>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>  In this context, the analogues of circular cones are not usually special; in fact one is often interested in <a href="/facts/Convex_cone/h1tARfw4">polyhedral cones</a>.
</p><p>An even more general concept is the <a href="/facts/Topological_cone/nUGIb0uE">topological cone</a>, which is defined in arbitrary topological spaces.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bicone/FabSqw34">Bicone</a></li>
<li><a href="/facts/Cone_(linear_algebra)/h1tARfw4">Cone (linear algebra)</a></li>
<li><a href="/facts/Cylinder_(geometry)/0wQPPr3k">Cylinder (geometry)</a></li>
<li><a href="/facts/Democritus/yTzsod3v">Democritus</a></li>
<li><a href="/facts/Elliptic_cone/n6h4xmLu">Elliptic cone</a></li>
<li><a href="/facts/Generalized_conic/6IIYaXyw">Generalized conic</a></li>
<li><a href="/facts/Hyperboloid/K5mDV2Nw">Hyperboloid</a></li>
<li><a href="/facts/List_of_shapes/lP1HBYsV">List of shapes</a></li>
<li><a href="/facts/Pyrometric_cone/Yo46ttGz">Pyrometric cone</a></li>
<li><a href="/facts/Quadric/L3cTApd4">Quadric</a></li>
<li><a href="/facts/Rotation_of_axes/lgzWdxpn">Rotation of axes</a></li>
<li><a href="/facts/Ruled_surface/uAGM9b1X">Ruled surface</a></li>
<li><a href="/facts/Translation_of_axes/3dbW7KB0">Translation of axes</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Protter, Murray H.; Morrey, Charles B. Jr. (1970), <i>College Calculus with Analytic Geometry</i> (2nd ed.), Reading: <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/76087042">76087042</a></li></ul>
<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Cones.

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Cone.html">"Cone"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/DoubleCone.html">"Double Cone"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/GeneralizedCone.html">"Generalized Cone"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li>An interactive <a href="http://www.mathsisfun.com/geometry/cone.html">Spinning Cone</a> from Maths Is Fun</li>
<li><a href="http://www.korthalsaltes.com/model.php?name_en=cone">Paper model cone</a></li>
<li><a href="http://mathforum.org/library/drmath/view/55017.html">Lateral surface area of an oblique cone</a></li>
<li><a href="http://www.cut-the-knot.org/Curriculum/Geometry/ConicSections.shtml">Cut a Cone</a> An interactive demonstration of the intersection of a cone with a plane</li></ul>

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

<ol>
<li id="fn:1"><p>James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410. <a href="9780412990410" target="_blank">9780412990410</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Grünbaum, Convex Polytopes, second edition, p. 23. <a href="/wiki/Convex_Polytopes" target="_blank">/wiki/Convex_Polytopes</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Weisstein, Eric W. "Cone". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410. <a href="9780412990410" target="_blank">9780412990410</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410. <a href="9780412990410" target="_blank">9780412990410</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Alexander, Daniel C.; Koeberlein, Geralyn M. (2014-01-01). Elementary Geometry for College Students. Cengage. ISBN 9781285965901. <a href="9781285965901" target="_blank">9781285965901</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Hartshorne, Robin (2013-11-11). Geometry: Euclid and Beyond. Springer Science & Business Media. Chapter 27. ISBN 9780387226767. <a href="9780387226767" target="_blank">9780387226767</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Blank, Brian E.; Krantz, Steven George (2006). Calculus: Single Variable. Springer. Chapter 8. ISBN 9781931914598. <a href="9781931914598" target="_blank">9781931914598</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Alexander, Daniel C.; Koeberlein, Geralyn M. (2014-01-01). Elementary Geometry for College Students. Cengage. ISBN 9781285965901. <a href="9781285965901" target="_blank">9781285965901</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Dowling, Linnaeus Wayland (1917-01-01). Projective Geometry. McGraw-Hill book Company, Incorporated. <a href="https://archive.org/details/projectivegeome04dowlgoog" target="_blank">https://archive.org/details/projectivegeome04dowlgoog</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>G. B. Halsted (1906) Synthetic Projective Geometry, page 20 <a href="/wiki/G._B._Halsted" target="_blank">/wiki/G._B._Halsted</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Grünbaum, Convex Polytopes, second edition, p. 23. <a href="/wiki/Convex_Polytopes" target="_blank">/wiki/Convex_Polytopes</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
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