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Geometry, coordinates, surveying and trigonometry
Volume of a prism
Volume of a prism from the area of its uniform cross-section and the perpendicular distance between its parallel ends.
- Ref
- GEO-16
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- V [cm³]
A prism has the same cross-section along its whole length. Stacking thin slices of area A over a perpendicular length h gives V = Ah. It holds for any cross-section shape, including an I-beam, a hexagon or a shape with holes, provided the ends are parallel and h is measured at right angles to them.
Volume of a prism from the area of its uniform cross-section and the perpendicular distance between its parallel ends. Any cross-section shape is allowed.
Calculator
- Volume in litres0.084 L
- Mass if the material has density 7.85 g/cm³0.6594 kg
- Volume if both dimensions double336 cm³
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A section of 12 cm² along 7 cm holds 84 cm³, which is 0.084 litres.
- If the same bar is steel (7.85 g/cm³) it weighs 0.6594 kg.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Cross-section area | 1 to 100 cm² | 7 | 700 | 693 | +10% | +10 % |
| h Length between the ends | 0.5 to 30 cm | 6 | 360 | 354 | +10% | +10 % |
Sweep of Length between the ends
| Length between the ends (cm) | Volume (cm³) |
|---|---|
| 0.5 | 6 |
| 4.188 | 50.25 |
| 7.875 | 94.5 |
| 11.56 | 138.8 |
| 15.25 | 183 |
| 18.94 | 227.3 |
| 22.63 | 271.5 |
| 26.31 | 315.8 |
| 30 | 360 |
Sweep one variable
Variables and how each one contributes
Volume
What the equation gives you. Everything else feeds it.
Cross-section area
The area of the uniform cross-section, in cm². A larger section gives a larger volume in direct proportion. Raising it by 10 % (to 13.2) moves the result by +10 %, from 84 to 92.4 cm³.
Length between the ends
The perpendicular distance between the two parallel ends, in cm. A longer prism gives a larger volume in direct proportion. Raising it by 10 % (to 7.7) moves the result by +10 %, from 84 to 92.4 cm³.
Worked example with the values above
- Cross-section area A = 12 cm² and length h = 7 cm.
- Multiply: 12 × 7.
- V = 84 cm³.
Assumptions and limits
- Solids with flat or smooth faces and a uniform cross-section where stated.
- All lengths in the same unit, with areas and volumes in the matching square and cubic units.
Common mistakes
- Using the slant height in place of the perpendicular height.
- Forgetting one end face in a closed prism, or counting an end that is open.
- Mixing cm² with cm³, or mm with cm, in one formula.
Statistics and uncertainty
Volumes scale with the cube of a length and surfaces with the square, so a 1 % error in a length gives about 3 % in a volume and 2 % in a surface.
Where it is used
- Material, paint and coating quantities for bars, tanks, hoppers and domes.
- Estimating volumes, masses and capacities from drawings.
Maths you need
- Volume: the space inside a solid, in cubic units
- Cross-section: the shape you see cutting the solid
- Surface area: the total area of all outer faces