Volume Formulas for Every Shape
The master table below lists the volume formula for every common solid. If you only remember two things: a shape with a constant cross-section holds base area × length, and a shape that tapers to a point holds one third of that. Every formula here is a special case of one of those, except the sphere.
The master formula table
| Shape | Volume formula | Symbols |
|---|---|---|
| Cube | a³ | a = edge length |
| Rectangular prism (box) | l × w × h | length, width, height |
| Cylinder | π r² h | r = radius, h = height |
| Tube (hollow cylinder) | π (R² − r²) h | R outer, r inner radius |
| Cone | ⅓ π r² h | r = base radius |
| Cone frustum | ⅓ π h (R² + Rr + r²) | R, r = end radii |
| Sphere | (4/3) π r³ | r = radius |
| Hemisphere | (2/3) π r³ | r = radius |
| Ellipsoid | (4/3) π a b c | a, b, c = semi-axes |
| Capsule | π r² (4/3 r + h) | r = radius, h = cylinder length |
| Spherical cap | (1/3) π h² (3R − h) | R = sphere radius, h = cap height |
| Truncated pyramid | (h/3)(A₁ + A₂ + √(A₁A₂)) | A₁, A₂ = parallel face areas |
| Pyramid (any base) | ⅓ × base area × h | h = perpendicular height |
| Square pyramid | ⅓ a² h | a = base edge |
| Triangular prism | ½ b ht × L | triangle base, height, length |
| Trapezoidal prism | ½ (a + b) ht × L | parallel sides, height, length |
| Prism (any base) | base area × L | L = length |
Round shapes explained
Cylinder — V = πr²h
A cylinder is a circle extruded along its length. Take the area of the circular end, πr², and multiply by the height. Example: a can 3 cm in radius and 10 cm tall holds π × 3² × 10 = 282.7 cm³ ≈ 283 mL.
Sphere — V = (4/3)πr³
The cube of the radius is what makes spheres grow so fast: double the radius and the volume grows eightfold. Example: a ball of radius 6 cm holds (4/3)π × 6³ = 904.8 cm³.
Cone — V = ⅓πr²h
Exactly one third of the cylinder with the same base and height. Example: a cone 4 cm across the base (r = 2) and 9 cm tall holds ⅓ × π × 2² × 9 = 37.7 cm³.
Flat-sided shapes explained
Box — V = l × w × h
The simplest formula there is: multiply the three edge lengths. Example: a 20 × 15 × 10 cm box holds 3,000 cm³ = 3 litres.
Prism — base area × length
Any prism — triangular, hexagonal, L-shaped — is its cross-section area times its length. Work out the area of the end face, then multiply. This single idea replaces a dozen separate formulas.
Pyramid — ⅓ × base area × height
Like the cone, a pyramid of any base holds one third of the matching prism. Use the perpendicular height from base to apex, not the slanted edge.
Where the "one third" comes from
The factor of ⅓ in every cone and pyramid is not arbitrary. Using calculus, the volume of a shape that scales linearly from a point is the integral of its cross-sectional area, which for a linear taper works out to exactly one third of the full prism. You can confirm it physically: fill a conical cup and pour it into a cylinder of the same base and height — it takes precisely three pours. The same holds for a pyramid inside its box.
A note on units
Every formula returns a volume in the cube of whatever length unit you put in. Measure in centimetres, get cubic centimetres (= millilitres). Mix units and the answer is meaningless, so convert first. Move between cubic and liquid measures with the conversion calculator, and see cubic units explained for why a cubic metre is a thousand litres.
Frequently asked questions
What is the general formula for volume?
There is no single formula for every shape, but two cover most: a shape with a constant cross-section has volume equal to its base area times its length, and a shape that tapers to a point holds one third of that. The sphere is the main exception at (4/3)πr³.
What is the formula for the volume of a cylinder?
V = π r² h, where r is the radius of the circular end and h is the height or length. It is the area of the circle multiplied by how long the cylinder is.
Why is the cone volume one third of the cylinder?
Any shape that narrows uniformly from a full base to a single point encloses exactly one third of the prism or cylinder with the same base and height. It is an exact geometric result, provable by calculus or by pouring water three times.
What is the formula for the volume of a sphere?
V = (4/3)π r³. A sphere occupies exactly two thirds of the cylinder that would just enclose it, a result first proved by Archimedes.