Volume formulas::Volume

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Volume formulas

Shape Volume formula Variables
Cube $a^3\;$ a = length of any side (or edge)
Cylinder $\pi r^2 h\;$ r = radius of circular face, h = height
Prism $B \cdot h$ B = area of the base, h = height
Rectangular prism $l \cdot w \cdot h$ l = length, w = width, h = height
Triangular prism $\frac{1}{2}bhl$ b = base length of triangle, h = height of triangle, l = length of prism or distance between the triangular bases
Sphere $\frac{4}{3} \pi r^3$ r = radius of sphere
which is the integral of the surface area of a sphere
Ellipsoid $\frac{4}{3} \pi abc$ a, b, c = semi-axes of ellipsoid
Torus $(\pi r^2)(2\pi R) = 2\pi^2 Rr^2$ r = minor radius (radius of the tube), R = major radius (distance from center of tube to center of torus)
Pyramid $\frac{1}{3}Bh$ B = area of the base, h = height of pyramid
Square pyramid $\frac{1}{3} s^2 h\;$ s = side length of base, h = height
Rectangular pyramid $\frac{1}{3} lwh$ l = length, w = width, h = height
Cone $\frac{1}{3} \pi r^2 h$ r = radius of circle at base, h = distance from base to tip or height
Tetrahedron<ref name=Cox>Coxeter, H. S. M.: Regular Polytopes (Methuen and Co., 1948). Table I(i).</ref> ${\sqrt{2}\over12}a^3 \,$ edge length $a$
Parallelepiped $a b c \sqrt{K}$
\begin{align} K =& 1+2\cos(\alpha)\cos(\beta)\cos(\gamma) \\ & - \cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)  \end{align}

a, b, and c are the parallelepiped edge lengths, and α, β, and γ are the internal angles between the edges
Any volumetric sweep
(calculus required)
$\int_a^b A(h) \,\mathrm{d}h$ h = any dimension of the figure,
A(h) = area of the cross-sections perpendicular to h described as a function of the position along h. a and b are the limits of integration for the volumetric sweep.
(This will work for any figure if its cross-sectional area can be determined from h).
Any rotated figure (washer method)
(calculus required)
$\pi \int_a^b \left({\left[R_O(x)\right]}^2 - {\left[R_I(x)\right]}^2\right) \mathrm{d}x$ $R_O$ and $R_I$ are functions expressing the outer and inner radii of the function, respectively.

Volume ratios for a cone, sphere and cylinder of the same radius and height

A cone, sphere and cylinder of radius r and height h

The above formulas can be used to show that the volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3, as follows.

Let the radius be r and the height be h (which is 2r for the sphere), then the volume of cone is

$\tfrac{1}{3} \pi r^2 h = \tfrac{1}{3} \pi r^2 (2r) = (\tfrac{2}{3} \pi r^3) \times 1,$

the volume of the sphere is

$\tfrac{4}{3} \pi r^3 = (\tfrac{2}{3} \pi r^3) \times 2,$

while the volume of the cylinder is

$\pi r^2 h = \pi r^2 (2r) = (\tfrac{2}{3} \pi r^3) \times 3.$

The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to Archimedes.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Volume sections
Intro  Units   Related terms    Volume in calculus    Volume formulas    Volume formula derivations    Volume in differential geometry    Volume in thermodynamics    See also   References  External links

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