Explanation
Measuring volume
Volume measures three-dimensional space enclosed by a solid. Enter all linear dimensions in metres. Answers are in cubic metres (m³); this calculator covers a rectangular prism, a right circular cylinder, and a sphere.
V = w × h × d
| V | volume in cubic metres (cubic metres (m³)) |
|---|---|
| w | prism width in metres (metres (m)) |
| h | height in metres (metres (m)) |
| d | prism depth in metres (metres (m)) |
V = πr²h
| V | volume in cubic metres (cubic metres (m³)) |
|---|---|
| r | radius in metres, from the axis or centre to the curved surface (metres (m)) |
| h | height in metres (metres (m)) |
V = 4πr³ ÷ 3
| V | volume in cubic metres (cubic metres (m³)) |
|---|---|
| r | radius in metres, from the axis or centre to the curved surface (metres (m)) |
A 2 m × 3 m × 4 m rectangular prism
| Width | 2 |
|---|---|
| Height | 3 |
| Depth | 4 |
| Result | 24 m³ |
2 m × 3 m × 4 m = 24 m³
A cylinder of radius 2 m and height 3 m
| Radius | 2 |
|---|---|
| Height | 3 |
| Result | 37.7 m³ |
π × (2 m)² × 3 m = 37.6991118431 m³
A sphere of radius 3 m
| Radius | 3 |
|---|---|
| Result | 113.1 m³ |
4π × (3 m)³ ÷ 3 = 113.0973355292 m³
Diagram of this result
The selected solid is drawn in the proportions you entered, with each measurement on a dimension line. The shaded body is the volume.
The entered measurements and the volume are stated in the answer and the steps.
Common mistakes
Use radius, not diameter, for the cylinder and sphere; the formulas square the radius and the sphere also cubes it. Do not omit a prism's depth. Keep units consistent—mixing centimetres and metres changes the numerical product and requires converting first. Volume has cubic units, not square units.
Assumptions and limits
Every required dimension must be positive and finite. Inputs are canonical decimals with the shared 30-significant-digit and adjusted-exponent limits and are interpreted as metres. The engine uses 50 significant digits and does not round intermediate products for display. The sphere and cylinder use π truncated after 50 decimal places; their answers are approximations with a comparison tolerance of 1e-12 relative plus 1e-12 absolute at the reference scale. Results beyond the shared output range are rejected.
Why are answers in cubic metres?
Multiplying three lengths multiplies their units three times, producing m × m × m = m³. Circle-based solids also combine a cross-sectional area in m² with a length in m.
Why is radius squared in the cylinder formula?
A cylinder's volume is its circular base area, πr², multiplied by its perpendicular height. The radius appears twice because it determines the base area.
References
For standard solid-volume formulas and measurement context, see OpenStax Prealgebra 2e, measurement and geometry topics: https://openstax.org/details/books/prealgebra-2e. The formulas are standard geometric identities; examples are independently checked against those identities.