This is a short list of some common mathematical shapes and figures and the formulas that describe them.
Two-dimensional shapes
| Shape | Area | Perimeter/Circumference | Meanings of symbols |
|---|---|---|---|
| Square | l 2 {\displaystyle l^{2}} | 4 l {\displaystyle 4l} | l {\displaystyle l} is the length of a side |
| Rectangle | l b {\displaystyle lb} | 2 ( l + b ) {\displaystyle 2(l+b)} | l {\displaystyle l} is length, b {\displaystyle b} is breadth |
| Circle | π r 2 {\displaystyle \pi r^{2}} | 2 π r {\displaystyle 2\pi r} or π d {\displaystyle \pi d} | where r {\displaystyle r} is the radius and d {\displaystyle d} is the diameter |
| Ellipse | π a b {\displaystyle \pi ab} | where a {\displaystyle a} is the semimajor axis and b {\displaystyle b} is the semiminor axis | |
| Triangle | b h 2 {\displaystyle {\frac {bh}{2}}} | a + b + c {\displaystyle a+b+c} | b {\displaystyle b} is base; h {\displaystyle h} is height; a , b , c {\displaystyle a,b,c} are sides |
| Parallelogram | b h {\displaystyle bh} | 2 ( a + b ) {\displaystyle 2(a+b)} | b {\displaystyle b} is base, h {\displaystyle h} is height, a {\displaystyle a} is side |
| Trapezoid | a + b 2 h {\displaystyle {\frac {a+b}{2}}h} | a {\displaystyle a} and b {\displaystyle b} are the bases |
Three-dimensional shapes
This is a list of volume formulas of basic shapes:4: 405–406
- Cone – 1 3 π r 2 h {\textstyle {\frac {1}{3}}\pi r^{2}h} , where r {\textstyle r} is the base's radius
- Cube – a 3 {\textstyle a^{3}} , where a {\textstyle a} is the side's length;
- Cuboid – a b c {\textstyle abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the sides' length;
- Cylinder – π r 2 h {\textstyle \pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cone's height;
- Ellipsoid – 4 3 π a b c {\textstyle {\frac {4}{3}}\pi abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the semi-major and semi-minor axes' length;
- Sphere – 4 3 π r 3 {\textstyle {\frac {4}{3}}\pi r^{3}} , where r {\textstyle r} is the radius;
- Parallelepiped – a b c K {\textstyle abc{\sqrt {K}}} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the sides' length, K = 1 + 2 cos ( α ) cos ( β ) cos ( γ ) − cos 2 ( α ) − cos 2 ( β ) − cos 2 ( γ ) {\textstyle K=1+2\cos(\alpha )\cos(\beta )\cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )} , and α {\textstyle \alpha } , β {\textstyle \beta } , and γ {\textstyle \gamma } are angles between the two sides;
- Prism – B h {\textstyle Bh} , where B {\textstyle B} is the base's area and h {\textstyle h} is the prism's height;
- Pyramid – 1 3 B h {\textstyle {\frac {1}{3}}Bh} , where B {\textstyle B} is the base's area and h {\textstyle h} is the pyramid's height;
- Tetrahedron – 2 12 a 3 {\textstyle {{\sqrt {2}} \over 12}a^{3}} , where a {\textstyle a} is the side's length.
Sphere
See also: Volume of an n-ball and n-sphere § Volume and surface area
The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables
- r {\displaystyle r} is the radius,
- C = 2 π r {\displaystyle C=2\pi r} is the circumference (the length of any one of its great circles),
- S {\displaystyle S} is the surface area,
- V {\displaystyle V} is the volume.
Surface area:
S = 4 π r 2 = 1 π C 2 = π ( 6 V ) 2 3 {\displaystyle {\begin{alignedat}{4}S&=4\pi r^{2}\\[0.3ex]&={\frac {1}{\pi }}C^{2}\\[0.3ex]&={\sqrt[{3}]{\pi (6V)^{2}}}\\[0.3ex]\end{alignedat}}}
Volume:
V = 4 3 π r 3 = 1 6 π 2 C 3 = 1 6 π S 3 / 2 {\displaystyle {\begin{alignedat}{4}V&={\frac {4}{3}}\pi r^{3}\\[0.3ex]&={\frac {1}{6\pi ^{2}}}C^{3}\\[0.3ex]&={\frac {1}{6{\sqrt {\pi }}}}S^{3/2}\\[0.3ex]\end{alignedat}}}
Radius:
r = 1 2 π C = 1 4 π S = 3 4 π V 3 {\displaystyle {\begin{alignedat}{4}r&={\frac {1}{2\pi }}C\\[0.3ex]&={\sqrt {{\frac {1}{4\pi }}S}}\\[0.3ex]&={\sqrt[{3}]{{\frac {3}{4\pi }}V}}\\[0.3ex]\end{alignedat}}}
C = 2 π r = π S = π 2 6 V 3 {\displaystyle {\begin{alignedat}{4}C&=2\pi r\\[0.3ex]&={\sqrt {\pi S}}\\[0.3ex]&={\sqrt[{3}]{\pi ^{2}6V}}\\[0.3ex]\end{alignedat}}}
See also
- Arc length – Distance along a curve
- Area#Area formulas – Size of a two-dimensional surface
- Perimeter#Formulas – Path that surrounds an area
- List of second moments of area
- List of surface-area-to-volume ratios – Surface area per unit volume
- List of surface area formulas – Measure of a two-dimensional surface
- List of trigonometric identities
- List of volume formulas – Quantity of three-dimensional space
References
"Archived copy" (PDF). Archived from the original (PDF) on 2012-08-13. Retrieved 2011-11-29.{{cite web}}: CS1 maint: archived copy as title (link) https://web.archive.org/web/20120813015606/http://www.austincc.edu/tutor/students/resources/Geometry.pdf ↩
"Area Formulas". http://www.math.com/tables/geometry/areas.htm ↩
"List of Basic Geometry Formulas". 27 May 2018. https://www.andlearning.org/geometry-formulas/ ↩
Treese, Steven A. (2018). History and Measurement of the Base and Derived Units. Cham, Switzerland: Springer Science+Business Media. ISBN 978-3-319-77577-7. LCCN 2018940415. OCLC 1036766223. 978-3-319-77577-7 ↩