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Volume Calculator: Volume of 3D Shapes
Choose a solid, enter its dimensions and get the volume with the formula worked out. For length units the result is also shown in litres and US gallons, handy for tanks, containers and pools.
Volume formulas used on this page
| Solid | Formula |
|---|---|
| Cube | a³ |
| Box | length × width × height |
| Cylinder | π r² h |
| Sphere | (4/3) π r³ |
| Hemisphere | (2/3) π r³ |
| Cone | (1/3) π r² h |
| Rectangular pyramid | (1/3) × length × width × height |
| Cone frustum | (π h ÷ 3)(R² + R r + r²) |
| Ellipsoid | (4/3) π a b c |
Volume counts how many unit cubes fit inside. The unit is therefore cubed: a length in centimetres gives cubic centimetres.
Patterns that make the formulas easy to remember
Most of these formulas follow two ideas. A prism or cylinder, which has the same cross-section all the way up, has volume equal to base area × height. A cone or pyramid that tapers to a point has exactly one third of the volume of the prism with the same base and height. A hemisphere is half of a sphere, and a sphere is a special ellipsoid with three equal semi-axes, so setting a, b and c all equal to r in the ellipsoid formula returns the sphere formula. A cone frustum is a cone with its tip cut off, and its formula reduces to the cone formula if the top radius is zero and to the cylinder formula if both radii are equal.
From cubic units to litres and gallons
One litre is exactly 1,000 cubic centimetres, which is why a 10 cm cube holds 1 litre. One US gallon is defined as 3.785411784 litres. For length units, the page converts your result to cubic metres, litres and US gallons so a tank or container size is easy to read. Imperial gallons are larger, at 4.54609 litres, so check which gallon a product label uses. Remember that interior dimensions determine capacity: a container with thick walls holds less than its outside measurements suggest.
Common mistakes
- Using diameter as radius. The radius is half the diameter, and because it is squared or cubed an error here multiplies quickly. Doubling the radius of a sphere multiplies its volume by eight.
- Forgetting the one third. Cones and pyramids hold a third of the matching cylinder or prism.
- Using slant height. The height in volume formulas is the perpendicular height from base to tip or top, not the length along a sloping side.
- Mixing units. Convert every dimension to the same unit before calculating, then convert the cubed result once at the end.
- Ignoring partial fill. A tank filled to a given depth holds the volume of a shorter solid, so enter the depth as the height.
Worked example
A cylindrical water tank has a radius of 1.5 m and a height of 2 m. How much does it hold?
Inputs
| Solid | Cylinder |
|---|---|
| Length unit | m |
| Height | 2 |
| Radius (bottom radius for a frustum) | 1.5 |
Result
Volume is the space inside the solid, in cubic units of the length unit you chose.
| Working | π × r² × h = π × 1.5² × 2 |
|---|---|
| Capacity in litres | 14,137.1669 L |
| Capacity in US gallons | 3,734.6444 gal |
| Volume in cubic metres | 14.1372 m³ |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
How do I find the volume of a cylinder?
Square the radius, multiply by pi and then by the height. A cylinder with radius 4 and height 8 has volume π × 16 × 8, which is about 402.12 cubic units.
Why is the volume of a cone one third of a cylinder?
A cone fills exactly one third of a cylinder with the same base and height. This ratio can be shown with calculus or by pouring experiments with matching shapes.
How many litres are in a cubic metre?
There are exactly 1,000 litres in one cubic metre, because a litre is 1,000 cubic centimetres and a metre is 100 centimetres.
What is the difference between volume and capacity?
Volume is the space the solid occupies, while capacity is how much a hollow container can hold. They match when you use the inside dimensions.
Can this calculator handle half-filled tanks?
Yes, indirectly. For an upright cylinder, enter the liquid depth as the height. Horizontal tanks and irregular shapes need different formulas.
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References and sources
- Standard solid geometry volume formulas; NIST Guide to the SI for litre and cubic metre definitions; US gallon defined as 3.785411784 litres.