This is a short list of some common mathematical shapes and figures and the formulas that describe them.
Two-dimensional shapes
Sources:
Three-dimensional shapes
This is a list of volume formulas of basic shapes:
Cone – 1 3 π r 2 h {\textstyle {\frac {1}{3}}\pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cone's height; 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 cylinder'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
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}}}
Circumference:
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 a three-dimensional space
References