Volume formulas
Cube, cylinder, sphere, cone: the volume formulas for the most common solids, ready to use.
Common solids
| Solid | Volume |
|---|---|
| Cube | side³ |
| Rectangular prism | length × width × height |
| Cylinder | π × radius² × height |
| Sphere | (4/3) × π × radius³ |
| Cone | (1/3) × π × radius² × height |
| Pyramid | (1/3) × base area × height |
A cone's or pyramid's volume is always a third of a cylinder's/prism's with the same base and height — a handy way to sanity-check a calculation.
Surface area (for reference)
| Solid | Total surface area |
|---|---|
| Cube | 6 × side² |
| Sphere | 4 × π × radius² |
| Cylinder | 2 × π × radius × (radius + height) |
Units: converting correctly
1 m³ = 1000 litres = 1,000,000 cm³ 1 litre = 1 dm³ = 1000 cm³ (or 1000 ml)
Surface area: cone and pyramid
Cone lateral area = π × radius × slant height Cone total area = π × radius × (radius + slant height) Pyramid lateral area = sum of the triangular faces' areas
A cone's slant height is the distance from the apex to a point on the base circle (not the height, which is perpendicular to the base): slant² = radius² + height² (Pythagorean theorem).
A worked example: the volume of a tank
A cylindrical tank with a 1.5 m radius and 4 m height:
Volume = π × 1.5² × 4 ≈ 28.27 m³ That's about 28,270 litres (1 m³ = 1000 L)
Comparing two solids of equal volume
A useful fact: at equal volume, the sphere is the solid that minimises surface area — which is why soap bubbles and water droplets in zero gravity naturally take a spherical shape (nature minimises surface energy).
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