Volume formulas::Volume
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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>
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
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
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