Library / Geometry, coordinates, surveying and trigonometry / GEO-18
Geometry, coordinates, surveying and trigonometry
Volume of a pyramid
Volume of a pyramid from its base area and the perpendicular height of the apex above the base plane.
- Ref
- GEO-18
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- V [cm³]
A pyramid has one third of the volume of the prism with the same base and height. The base can be any polygon and h is the perpendicular height from the apex to the base plane, not the slant height. For the volume of a cone use the base area πr² in the same formula.
Volume of a pyramid from its base area and the perpendicular height of the apex above the base plane. The base can be any polygon, and the slant height is not used.
Calculator
- Volume of the prism with the same base and height144 cm³
- Base side if the base is a square6 cm
- Volume in litres0.048 L
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 pyramid on a 36 cm² base, 4 cm tall, holds 48 cm³, one third of the 144 cm³ prism round it.
- Three such pyramids fill the prism, which is the standard cube-split demonstration.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Perpendicular height | 0.5 to 30 cm | 6 | 360 | 354 | +10% | +10 % |
| A Base area | 1 to 200 cm² | 1.333 | 266.7 | 265.3 | +10% | +10 % |
Sweep of Perpendicular height
| Perpendicular height (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.
Base area
The area of the base, in cm². A bigger base gives a larger volume in direct proportion. Raising it by 10 % (to 39.6) moves the result by +10 %, from 48 to 52.8 cm³.
Perpendicular height
The perpendicular height from the apex to the plane of the base, in cm. A taller pyramid gives a larger volume in direct proportion. Raising it by 10 % (to 4.4) moves the result by +10 %, from 48 to 52.8 cm³.
Worked example with the values above
- Base area A = 36 cm² and height h = 4 cm.
- Multiply: 36 × 4 = 144.
- V = 144 / 3 = 48 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