Volume - Volume Formulas

Volume Formulas

Shape Volume formula Variables
Cube a = length of any side (or edge)
Cylinder r = radius of circular face, h = height
Prism B = area of the base, h = height
Rectangular prism l = length, w = width, h = height
Sphere r = radius of sphere
which is the integral of the surface area of a sphere
Ellipsoid a, b, c = semi-axes of ellipsoid
Pyramid B = area of the base, h = height of pyramid
Cone r = radius of circle at base, h = distance from base to tip or height
Tetrahedron edge length
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)		 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)		 and are functions expressing the outer and inner radii of the function, respectively.
Klein bottle No volume—it has no inside.

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