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Volume formulas

Shape Volume formula Variables
Cube <math>a^3\;</math> a = length of any side (or edge)
Cylinder <math>\pi r^2 h\;</math> r = radius of circular face, h = height
Prism <math>B \cdot h</math> B = area of the base, h = height
Rectangular prism <math>l \cdot w \cdot h</math> l = length, w = width, h = height
Triangular prism <math>\frac{1}{2}bhl</math> b = base length of triangle, h = height of triangle, l = length of prism or distance between the triangular bases
Sphere <math>\frac{4}{3} \pi r^3</math> r = radius of sphere
which is the integral of the surface area of a sphere
Ellipsoid <math>\frac{4}{3} \pi abc</math> a, b, c = semi-axes of ellipsoid
Torus <math>(\pi r^2)(2\pi R) = 2\pi^2 Rr^2</math> r = minor radius (radius of the tube), R = major radius (distance from center of tube to center of torus)
Pyramid <math>\frac{1}{3}Bh</math> B = area of the base, h = height of pyramid
Square pyramid <math>\frac{1}{3} s^2 h\;</math> s = side length of base, h = height
Rectangular pyramid <math>\frac{1}{3} lwh</math> l = length, w = width, h = height
Cone <math>\frac{1}{3} \pi r^2 h</math> 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> <math>{\sqrt{2}\over12}a^3 \,</math> edge length <math>a</math>
Parallelepiped <math>

a b c \sqrt{K} </math>
<math> \begin{align}

K =& 1+2\cos(\alpha)\cos(\beta)\cos(\gamma) \\
& - \cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)

\end{align} </math>

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)
<math>\int_a^b A(h) \,\mathrm{d}h</math> 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)
<math>\pi \int_a^b \left({\left[R_O(x)\right]}^2 - {\left[R_I(x)\right]}^2\right) \mathrm{d}x</math> <math>R_O</math> and <math>R_I</math> 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

<math>\tfrac{1}{3} \pi r^2 h = \tfrac{1}{3} \pi r^2 (2r) = (\tfrac{2}{3} \pi r^3) \times 1,</math>

the volume of the sphere is

<math>\tfrac{4}{3} \pi r^3 = (\tfrac{2}{3} \pi r^3) \times 2,</math>

while the volume of the cylinder is

<math>\pi r^2 h = \pi r^2 (2r) = (\tfrac{2}{3} \pi r^3) \times 3.</math>

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  

Volume formulas
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