Volume Formulas

Volume formulas give the amount of space a three-dimensional solid occupies, always in cubic units. The essentials are cube V = s³, cylinder V = πr²h, cone V = ⅓πr²h, and sphere V = 4⁄3·πr³. The table below covers every common solid, each linking to a live explorer.

All 3D volume & surface-area formulas

Solid	Volume	Surface area
Cube	$V = s^3$	$SA = 6s^2$
Rectangular Prism	$V = l w h$	$SA = 2(lw + lh + wh)$
Triangular Prism	$V = A_{base}\cdot L$	$SA = 2A_{base} + (a+b+c)L$
Cylinder	$V = \pi r^2 h$	$LSA = 2\pi r h$ $SA = 2\pi r(r + h)$
Cone	$V = \tfrac{1}{3}\pi r^2 h$	$LSA = \pi r \ell$ $SA = \pi r(r + \ell)$
Sphere	$V = \tfrac{4}{3}\pi r^3$	$SA = 4\pi r^2$
Hemisphere	$V = \tfrac{2}{3}\pi r^3$	$2\pi r^2$ $SA = 3\pi r^2$
Pyramid	$V = \tfrac{1}{3} b^2 h$	$LSA = 2 b \ell$ $SA = b^2 + 2 b \ell$

Worked examples

Cube side 2: V = s³ = 2³ = 8 cubic units.
Cylinder r = 3, h = 6: V = πr²h = π·9·6 = 54π ≈ 169.65.
Cone r = 3, h = 4: V = ⅓πr²h = ⅓·π·9·4 = 12π ≈ 37.70.
Sphere r = 3: V = 4⁄3·πr³ = 36π ≈ 113.10.

Variable glossary

a, b, cside lengths
l, w, hlength, width, height
rradius
ddiameter (d = 2r)
sside length or slant height
ℓslant height
bbase
Pperimeter
Ccircumference
Aarea
SAsurface area
Vvolume
θcentral angle (degrees)

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Frequently asked questions

What are the basic volume formulas?

The core volume formulas are: cube V = s³, rectangular prism V = l·w·h, cylinder V = πr²h, cone V = ⅓πr²h, sphere V = 4⁄3·πr³, and pyramid V = ⅓·(base area)·h. Volume is measured in cubic units.

Why is a cone one third of a cylinder?

A cone and a cylinder with the same base radius and height differ by exactly a factor of three: V_cone = ⅓πr²h while V_cylinder = πr²h. The same ⅓ relates a pyramid to its prism.

What is the volume of a sphere?

A sphere’s volume is V = 4⁄3·πr³. For radius 3, that is 4⁄3·π·27 = 36π ≈ 113.10 cubic units.