Library / Foundations: Maths / FMA-10
Foundations: Maths
Volume of a cone and cylinder
Volume from base radius and perpendicular height.
- Ref
- FMA-10
- Rev
- v1
- Field
- Foundations: Maths
- Result
- V [cm³]
A prism has volume V = Ah, with A the constant cross-section area. A cylinder is a prism with A = πr². A cone has one third of the cylinder volume.
A cone holds exactly one third of the cylinder with the same base and height.
Calculator
cm³
cm
cm
- Cylinder volume πr²h141.4 cm³
- Volume in litres0.04712 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.
Cone volume (cm³) current value 47.12 cm³
rBase radiusKey driver
1.314,710
hPerpendicular height
4.71471
- Use the perpendicular height, not the slant height.
- Radius is squared, so it dominates the volume.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Base radius | 0.5 to 30 cm | 1.309 | 4,712 | 4,711 | +10% | +21 % |
| h Perpendicular height | 0.5 to 50 cm | 4.712 | 471.2 | 466.5 | +10% | +10 % |
Sweep of Base radius
| Base radius (cm) | Cone volume (cm³) |
|---|---|
| 0.5 | 1.309 |
| 0.8341 | 3.643 |
| 1.392 | 10.14 |
| 2.322 | 28.22 |
| 3.873 | 78.54 |
| 6.461 | 218.6 |
| 10.78 | 608.4 |
| 17.98 | 1,693 |
| 30 | 4,712 |
Sweep one variable
Cone volume (cm³)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
V
Cone volume
What the equation gives you. Everything else feeds it.
r
Base radius
Radius of the circular base. Squared: +10% radius gives +21% volume.
h
Perpendicular height
Height measured at right angles to the base. Directly proportional.
Worked example with the values above
- Base area: π × 3² = 28.27 cm².
- Cylinder volume: 28.27 × 5 = 141.4 cm³.
- One third: 47.12 cm³.
Assumptions and limits
- A right circular cone.
- Height perpendicular to the base.
Common mistakes
- Using the slant height.
- Forgetting the one third.
Statistics and uncertainty
Relative errors combine as δV/V ≈ √[(2δr/r)² + (δh/h)²]; the radius counts twice.
Where it is used
- Volume of piles, funnels and nose cones.
- Tank and hopper capacities.
Maths you need
- Circle area: πr²
- Fractions: one third