Library / Geometry, coordinates, surveying and trigonometry / GEO-20
Geometry, coordinates, surveying and trigonometry
Volume of a conical frustum
Volume of a cone with its top cut off by a plane parallel to the base, from the two end radii and the perpendicular height.
- Ref
- GEO-20
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- V [cm³]
A frustum is a cone with its top cut off by a plane parallel to the base. With end radii R and r and perpendicular height h its volume is πh(R² + Rr + r²)/3. It equals the volume of the big cone minus the small cone removed. When r = 0 it becomes a cone, and when R = r it becomes a cylinder.
Volume of a cone with its top cut off by a plane parallel to the base, from the two end radii and the perpendicular height. At R = r it gives a cylinder.
Calculator
- Volume of a cylinder of the larger radius150.8 cm³
- Share of that cylinder0.5833
- Slant height, √(h² + (R − r)²)3.606 cm
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 frustum with end radii 4 and 2 cm and height 3 cm holds 87.96 cm³, 58.3 % of the cylinder of the large radius.
- Setting the small radius to 4 cm would give a cylinder of 150.8 cm³.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm³) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Larger end radius | 2 to 20 cm | 37.7 | 1,395 | 1,357 | +10% | +14.9 % |
| h Perpendicular height | 0.5 to 20 cm | 14.66 | 586.4 | 571.8 | +10% | +10 % |
| r Smaller end radius | 0 to 4 cm | 50.27 | 150.8 | 100.5 | +10% | +5.86 % |
Sweep of Smaller end radius
| Smaller end radius (cm) | Volume (cm³) |
|---|---|
| 0 | 50.27 |
| 0.5 | 57.33 |
| 1 | 65.97 |
| 1.5 | 76.18 |
| 2 | 87.96 |
| 2.5 | 101.3 |
| 3 | 116.2 |
| 3.5 | 132.7 |
| 4 | 150.8 |
Sweep one variable
Variables and how each one contributes
Volume
What the equation gives you. Everything else feeds it.
Larger end radius
The radius of the larger end, in cm. A larger end raises the volume, and it enters squared. Raising it by 10 % (to 4.4) moves the result by +14.9 %, from 87.96 to 101 cm³.
Smaller end radius
The radius of the smaller end, in cm. It can be zero. A larger small end raises the volume, but less than the large end. Raising it by 0.1 (to 2.1) moves the result by +2.89 %, from 87.96 to 90.51 cm³.
Perpendicular height
The perpendicular height between the two circular ends, in cm. A taller frustum gives a larger volume in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 87.96 to 96.76 cm³.
Worked example with the values above
- Bracket: R² + Rr + r² = 16 + 8 + 4 = 28.
- Multiply by πh: π × 3 × 28 = 263.89.
- V = that / 3 = 87.965 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.
- Taking the formula for a cone minus a cone with the wrong heights.
- Entering the smaller radius as larger than the bigger one.
Statistics and uncertainty
The volume depends on the squares of the two radii and their product, so errors in the radii matter more than errors in the height.
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