Volume Formulas for All 3D Shapes — Complete Reference
Volume measures how much 3D space a shape occupies — the number of unit cubes that fit inside it. Every 3D shape has a volume formula, and most can be derived from the basic principle that volume = base area × height (for prisms and cylinders) or (1/3) × base area × height (for pyramids and cones).
This guide covers volume formulas for all common 3D shapes, explains why each formula works, and includes a complete reference table with worked examples.
Formula
Cylinder: V = πr²h | Sphere: V = (4/3)πr³ | Cone: V = (1/3)πr²h
Volume Formula Reference Table
Use this table as a quick reference for all common 3D shape volume formulas.
| Shape | Formula | Variables | Example |
|---|---|---|---|
| Cube | V = s³ | s = side length | s=4 → V=64 |
| Rectangular Prism | V = l × w × h | l,w,h = dimensions | 5×4×3 = 60 |
| Cylinder | V = πr²h | r = radius, h = height | r=3,h=5 → V≈141.37 |
| Sphere | V = (4/3)πr³ | r = radius | r=6 → V≈904.78 |
| Cone | V = (1/3)πr²h | r = radius, h = height | r=4,h=9 → V≈150.80 |
| Square Pyramid | V = (1/3)b²h | b = base side, h = height | b=6,h=8 → V=96 |
| Triangular Prism | V = ½bhl | b,h = triangle base&height, l = length | ½×4×3×10=60 |
| Hemisphere | V = (2/3)πr³ | r = radius | r=5 → V≈261.80 |
| Ellipsoid | V = (4/3)πabc | a,b,c = semi-axes | a=3,b=2,c=1 → V≈25.13 |
| Frustum (cone) | V = (πh/3)(R²+Rr+r²) | R,r = radii, h = height | R=5,r=3,h=4 → V≈205.25 |
| Torus | V = 2π²Rr² | R = major radius, r = minor radius | R=4,r=1 → V≈78.96 |
| Triangular Pyramid | V = (1/3) × base area × h | base area = ½bw, h = height | base 12, h=5 → V=20 |
The Prism Principle — Base Area × Height
For any prism or cylinder (shapes with uniform cross-section), volume = base area × height. A prism has the same cross-sectional shape from bottom to top — whether rectangular, triangular, hexagonal, or circular (a cylinder).
Why? Imagine stacking very thin slices of the shape. Each slice has volume ≈ base area × slice thickness. Sum all slices: V = base area × total height.
Examples: Cube (square base s², height s): V = s² × s = s³. Cylinder (circle base πr², height h): V = πr² × h. Triangular prism (triangle base ½bw, length l): V = ½bwl.
Why Cones and Pyramids Are 1/3 of the Enclosing Prism
A cone has volume exactly (1/3) × πr²h — exactly one-third of the enclosing cylinder. A pyramid has volume (1/3) × base area × height — one-third of the enclosing prism.
This 1/3 factor is non-obvious but can be proved rigorously via calculus: a cone is formed by rotating a right triangle about the altitude axis, and integrating circular cross-sections of increasing radius from 0 to r as height increases from 0 to h.
More intuitively: fill a cone with water, pour it into a cylinder with the same base and height. It takes exactly 3 fills. This empirical approach is how ancient mathematicians first discovered the 1/3 rule.
Sphere Volume — Why (4/3)πr³?
Archimedes proved that a sphere of radius r has volume (4/3)πr³ by showing it equals (2/3) of the circumscribed cylinder (radius r, height 2r). The cylinder has volume π r² × 2r = 2πr³. So sphere = (2/3) × 2πr³ = (4/3)πr³.
The modern calculus derivation: integrate circular cross-sections. At height y from the center (−r ≤ y ≤ r), the circular cross-section has radius √(r²−y²) and area π(r²−y²). Integrating: V = ∫₋ᵣʳ π(r²−y²)dy = π[r²y − y³/3]₋ᵣʳ = (4/3)πr³.
Note: the surface area SA = 4πr² can be obtained by differentiating the volume with respect to r: dV/dr = d/dr[(4/3)πr³] = 4πr² = SA. This beautiful relationship means surface area is the "rate of change" of volume.
Volume Unit Conversions
Volume units are cubic length units. Converting can be tricky because the cube exponent applies to the unit conversion factor.
| From | To | Multiply by |
|---|---|---|
| cm³ | liters (L) | 0.001 |
| liters (L) | cm³ | 1000 |
| m³ | liters | 1000 |
| in³ | cm³ | 16.387 |
| ft³ | liters | 28.317 |
| US gallons | liters | 3.7854 |
| US gallons | in³ | 231 |
| UK (imperial) gallons | liters | 4.5461 |