Case 1
- Bottom radius
- 5
- Top radius
- 3
- Vertical height
- 10
513.126800
Open with these valuesFrustum Volume Calculator
Change any value to update the result.
513.126800units³
Volume: 513.126800 units³Examples
A frustum is a cone with its tip sliced off parallel to the base: a bucket, a plant pot, a lampshade. It is not the average of the two circles — the middle term R·r is what makes the sides straight instead of curved. Equal radii give back the cylinder.
The numbers at a glance
Held fixed: Bottom radius 5.000, Top radius 3.000.
| Vertical height | Volume |
|---|---|
| 2.500 | 128.281700 |
| 5.000 | 256.563400 |
| 7.500 | 384.845100 |
| 10.000Your value | 513.126800 |
| 12.500 | 641.408500 |
| 15.000 | 769.690200 |
| 17.500 | 897.971900 |
| 20.000 | 1,026.253600 |
Worked examples
Case 2
- Bottom radius
- 1
- Top radius
- 1
- Vertical height
- 1
3.141593
Open with these valuesCase 3
- Bottom radius
- 10
- Top radius
- 5
- Vertical height
- 20
3,665.191429
Open with these valuesHow it's calculated
V = ⅓ × π × h × (R² + R·r + r²)
- StepMeasure the radius of the wide end and of the narrow end.
- StepMeasure the height straight between the two circles.Vertical height=10
- ResultRead the volume in the cube of the unit you entered.513.126800units³
What this number means
Not the average of the two end circles
Averaging them gives 534.07 for the bucket of radii 5 and 3 over a height of 10, instead of 513.13 — about four per cent too much. The side is a straight slope, not a step.
The middle term R·r does the work
It is the only reason the bracket is not simply two circle areas added up. Set R = r and it turns into 3R², collapsing the whole formula back to the cylinder πR²h.
A bucket, a plant pot, a lampshade
Each of those needs three measurements: the radius of the wide opening, the radius of the narrow one, and the height straight between the two circles.
Commonly misread
MisconceptionI averaged the two circle areas and multiplied by the height.
CorrectionThat overstates the bucket of radii 5 and 3 by about four per cent, 534.07 instead of 513.13. Use R² + R·r + r² in the bracket.
MisconceptionI put the wider radius in the top field by mistake.
CorrectionNo harm done. The formula is symmetric in R and r, so a bucket upside down comes out the same.
MisconceptionBoth openings are the same size, so this is the wrong calculator.
CorrectionIt still works: equal radii turn the formula into πR²h, the cylinder. Entering 1, 1 and 1 returns π, which is a good check that the inputs landed where you meant.
Reference table
| Bottom, top, height | Note | Volume |
|---|---|---|
| 0.5, 0.25, 3 | small pot | 1.374447 |
| 1, 1, 1 | equal radii → cylinder π | 3.141593 |
| 2, 1, 6 | halves at each end | 43.982297 |
| 5, 3, 10 | typical bucket | 513.126800 |
| 10, 6, 20 | bucket doubled → eight times over | 4105.014401 |
Questions
How do you find the volume of a frustum?
One third of π times the height times R² + R·r + r². A bucket 10 units tall with radii 5 and 3 holds about 513.13 cubic units.
Why not just average the two circles?
Because the side is a straight slope, not a step. Averaging the end areas gives 534.07 for the bucket above instead of 513.13 — about four per cent too much.
Does it matter which radius I call the top?
No. The formula is symmetric in R and r, so swapping them returns the same volume. A bucket and the same bucket upside down hold the same amount.
What if both radii are equal?
Then it is a cylinder and the formula collapses to πR²h. That is a useful check: enter 1, 1 and 1 and you should read π.