Each common 3D shape has its own volume formula, but cylinders, cones and spheres share a surprisingly elegant relationship when they share the same radius and height.
The formulas
- Cube: side³
- Box: length × width × height
- Cylinder: π × radius² × height
- Sphere: (4/3) × π × radius³
- Cone: (π × radius² × height) ÷ 3
Worked example: the cylinder-cone-sphere relationship
For a radius of 3: a cylinder of height 6 has volume ≈ 169.646. A cone with the *same* radius and height has volume ≈ 56.549 — exactly one-third of the cylinder's volume, as the formula's ÷3 suggests directly. A sphere with that same radius (so its diameter equals the cylinder's height) has volume ≈ 113.097 — exactly two-thirds of the cylinder's volume. This 1:2:3 ratio (cone : sphere : cylinder) holds for any radius, as long as the cylinder's height equals its diameter.
Common mistakes
- Using the diameter instead of the radius in any of these formulas — since volume typically scales with radius cubed or squared, this error compounds quickly.
- Forgetting that cone and sphere volumes use the same radius definition as the cylinder they're being compared to — the elegant 1:2:3 ratio only holds when height equals diameter.
Calculate the volume of a cube, box, cylinder, sphere or cone with the Volume Calculator.
Related tools