Answer first
What this calculator tells you
Calculate the volume, slant height and surface area of a truncated cone from its two radii and height. Work out a bucket, a lampshade or a planter that is a cone with the tip cut off. Formula: Volume = π × h × (R² + R×r + r²) ÷ 3; slant = √(h² + (R − r)²); total surface = π × (R² + r²) + π × (R + r) × slant. At the worked-example inputs, the volume is 527.8. Holding every other input steady, moving bottom radius (r) from 4.8 to 7.2 moves the result from 389.1 to 690.6.
Transparent method
The formula
Work out a bucket, a lampshade or a planter that is a cone with the tip cut off.
Worked example
Example inputs
How to interpret the result
A frustum is a cone with the tip sliced off, so its volume is a difference of two cones folded into one formula. A bucket with a bottom radius of 6, a top radius of 3 and a height of 8 holds about 528 cubic units. The slant height, about 8.54, is the length of the sloping wall, and it is longer than the straight-up height whenever the radii differ.
At the worked-example inputs the volume is 527.8. It rises with bottom radius (r), height (h) and top radius (r).
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer.
Before you rely on it
What to check
Use the perpendicular height between the two circles, not the slanted side. The slanted side belongs only in the surface area.
The common error
Where people go wrong with cone frustum calculator
Averaging the two radii and using a cylinder formula. That works only for a very slight taper, and it misses the volume as the sides lean in.
Sensitivity evidence
How bottom radius (r) changes the volume
Holding every other input at the worked-example value, moving bottom radius (r) from 4.8 to 7.2 moves the volume from 389.1 to 690.6: a spread of 301.6, or 57% of the worked-example result.
| Bottom radius (R) | Volume | Slant height | Total surface area |
|---|---|---|---|
| 4.8 | 389.1 | 8.2 | 301.6 |
| 5.4 | 455.4 | 8.4 | 340.3 |
| 6worked example | 527.8 | 8.5 | 382.9 |
| 6.6 | 606.2 | 8.8 | 429.7 |
| 7.2 | 690.6 | 9 | 480.7 |
Every input, tested
Which input moves the volume most
Of the 3 inputs, bottom radius (r) moves the volume most (150.8 across the range tested) and top radius (r) moves it least (60.3).
| Input | Tested from | To | Volume at each end | Swing |
|---|---|---|---|---|
| Bottom radius (R) | 5.4 | 6.6 | 455.4 to 606.2 | 150.8 (29%) |
| Height (h) | 7 | 9 | 461.8 to 593.8 | 131.9 (25%) |
| Top radius (r) | 2.7 | 3.3 | 498.4 to 558.7 | 60.3 (11%) |
Two variables at once
Volume by bottom radius (r) and top radius (r)
Across the grid the volume runs from 337.8 to 760. Moving bottom radius (r) from 4.8 to 7.2 shifts it by 301.6 at the middle column, and moving top radius (r) from 2.4 to 3.6 shifts it by 120.6 at the middle row, so bottom radius (r) is the bigger lever here.
| Bottom radius (R) \ Top radius (r) | 2.4 | 3 | 3.6 |
|---|---|---|---|
| 4.8 | 337.8 | 389.1 | 446.4 |
| 5.4 | 401.1 | 455.4 | 515.7 |
| 6 | 470.5 | 527.8 | 591.1 |
| 6.6 | 545.9 | 606.2 | 672.6 |
| 7.2 | 627.3 | 690.6 | 760 |
The highlighted cell is the worked example: 527.8.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Bottom radius (R) | 6 | Enter the bottom radius (r) used in this calculation. |
| Top radius (r) | 3 | Enter the top radius (r) used in this calculation. |
| Height (h) | 8 | Enter the height (h) used in this calculation. |
| Volume | 527.8 | |
| Slant height | 8.5 | |
| Total surface area | 382.9 | |
Inputs, definitions and assumptions
Bottom radius (R)
Enter the bottom radius (r) used in this calculation. The prefilled worked-example value is 6.
Top radius (r)
Enter the top radius (r) used in this calculation. The prefilled worked-example value is 3.
Height (h)
Enter the height (h) used in this calculation. The prefilled worked-example value is 8.
How to use this calculator
- 1Verify the inputs. Gather bottom radius (r), top radius (r) and height (h) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the volume at 527.8. Store your own version of it as Scenario A.
- 3Test one change. Start with bottom radius (r), the input with the biggest effect here: moving bottom radius (r) from 5.4 to 6.6 takes the volume from 455.4 to 606.2, a swing of 29% of the worked-example figure.
- 4Check the extremes. At half the example bottom radius (r) (3) the volume is 226.2; at double (12) it is 1,583.4.
People also ask
Frequently asked questions
How do you calculate cone frustum?
Volume = π × h × (R² + R×r + r²) ÷ 3; slant = √(h² + (R − r)²); total surface = π × (R² + r²) + π × (R + r) × slant. At the worked-example inputs the volume is 527.8.
What does the cone frustum result mean?
Work out a bucket, a lampshade or a planter that is a cone with the tip cut off. At the worked-example inputs the volume is 527.8. It rises with bottom radius (r), height (h) and top radius (r).
How much does bottom radius (r) change the volume?
Holding every other input at the worked-example value, moving bottom radius (r) from 4.8 to 7.2 moves the volume from 389.1 to 690.6, a spread of 301.6.
What are the limits of this cone frustum calculator?
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer. The tables on this page test bottom radius (r) only from 4.8 to 7.2; a value outside that range is not tabulated here.
Which input moves the volume most in the cone frustum calculator?
Ranked by how far each moves the volume across the range tested: bottom radius (r) (150.8, 29%), height (h) (131.9, 25%) and top radius (r) (60.3, 11%).
If I double bottom radius (r) in the cone frustum calculator, does the volume double?
Doubling it from 6 to 12 takes the volume from 527.8 to 1,583.4, which is 3.00 times the worked-example figure. So the result grows faster than the input does. Halving it to 3 gives 226.2.
How much does top radius (r) matter in the cone frustum calculator?
The worked example uses 3. Holding every other input at its worked-example value, moving top radius (r) from 2.7 to 3.3 takes the volume from 498.4 to 558.7, a swing of 11% of the worked-example figure.
How much does height (h) matter in the cone frustum calculator?
The worked example uses 8. Holding every other input at its worked-example value, moving height (h) from 7 to 9 takes the volume from 461.8 to 593.8, a swing of 25% of the worked-example figure.
Which inputs change the slant height in the cone frustum calculator?
At the worked-example inputs it is 8.5. Bottom radius (r) takes it from 8.4 to 8.8, top radius (r) takes it from 8.7 to 8.4 and height (h) takes it from 7.6 to 9.5.
Which inputs change the total surface area in the cone frustum calculator?
At the worked-example inputs it is 382.9. Bottom radius (r) takes it from 340.3 to 429.7, top radius (r) takes it from 372.5 to 394 and height (h) takes it from 356.7 to 409.6.
What is the golden ratio and where does it show up?
It is about 1.618, the ratio at which the whole is to the longer part as the longer part is to the shorter. It appears in the Fibonacci sequence, in some plants and in design layouts, though many claims about it are exaggerated.
Sources and evidence
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