Frustum Volume Calculator (Frustum of a Cone)
Compute the volume, lateral surface area, and total surface area of a conical frustum from its radii and height. Supports metric and imperial units with step‑by‑step formulas.
Frustum of a Cone Calculator
Results
Enter all dimensions and click “Calculate”.
- Volume
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- Lateral surface area
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- Total surface area
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- Slant height s
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How to use the frustum volume calculator
- Select the unit system (metric or imperial).
- Enter the top radius \(r_1\) and bottom radius \(r_2\). Radii must be non‑negative.
- Enter the vertical height \(h\) of the frustum (not the slant height).
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Click Calculate to get:
- Volume \(V\)
- Lateral surface area \(A_L\)
- Total surface area \(A_T\)
- Slant height \(s\)
- Enable “Show detailed steps” to see the intermediate values and substituted formulas.
Frustum of a cone volume formula
A conical frustum is obtained by cutting a right circular cone with a plane parallel to its base and removing the top part. It has two parallel circular faces with radii \(r_1\) and \(r_2\) and vertical height \(h\).
Volume of a conical frustum
\[ V = \frac{1}{3}\,\pi\,h\,(r_1^2 + r_1 r_2 + r_2^2) \]
where:
- \(r_1\) = radius of the top circle
- \(r_2\) = radius of the bottom circle
- \(h\) = vertical height between the two circles
Derivation (idea)
Start with a full cone of height \(H\) and base radius \(R\). Slice it with a plane parallel to the base at height \(h\) from the base. The top removed cone is similar to the original cone, so the radii and heights are proportional. Subtracting the volume of the small cone from the big cone leads to the compact formula above.
Surface area formulas
First compute the slant height \(s\) using the Pythagorean theorem:
\[ s = \sqrt{(r_2 - r_1)^2 + h^2} \]
The lateral surface area (curved side) is:
\[ A_L = \pi\,(r_1 + r_2)\,s \]
The total surface area (including both circular ends) is:
\[ A_T = A_L + \pi r_1^2 + \pi r_2^2 \]
Worked example
Suppose you have a frustum with:
- Top radius \(r_1 = 4\ \text{cm}\)
- Bottom radius \(r_2 = 6\ \text{cm}\)
- Height \(h = 10\ \text{cm}\)
1. Volume
Use the volume formula:
\[ V = \frac{1}{3}\pi h (r_1^2 + r_1 r_2 + r_2^2) \]
Substitute the values:
\[ \begin{aligned} V &= \frac{1}{3}\pi \cdot 10 \left(4^2 + 4\cdot 6 + 6^2\right) \\ &= \frac{10}{3}\pi \left(16 + 24 + 36\right) \\ &= \frac{10}{3}\pi \cdot 76 \\ &= \frac{760}{3}\pi \approx 796.18\ \text{cm}^3 \end{aligned} \]
2. Slant height
\[ s = \sqrt{(r_2 - r_1)^2 + h^2} = \sqrt{(6 - 4)^2 + 10^2} = \sqrt{2^2 + 100} = \sqrt{104} \approx 10.20\ \text{cm} \]
3. Lateral and total surface area
Lateral area:
\[ A_L = \pi (r_1 + r_2) s = \pi (4 + 6)\cdot 10.20 = 10\pi \cdot 10.20 \approx 320.6\ \text{cm}^2 \]
Total area:
\[ A_T = A_L + \pi r_1^2 + \pi r_2^2 = 320.6 + \pi(4^2 + 6^2) = 320.6 + \pi(16 + 36) = 320.6 + 52\pi \approx 484.0\ \text{cm}^2 \]
FAQ about frustum volume
What is the formula for the volume of a frustum of a cone?
The volume of a right circular conical frustum is
\[ V = \frac{1}{3}\,\pi\,h\,(r_1^2 + r_1 r_2 + r_2^2) \]
where \(r_1\) and \(r_2\) are the radii of the two circular faces and \(h\) is the vertical height.
Can I use diameter instead of radius?
Yes. If you know the diameters \(D_1\) and \(D_2\), first convert them to radii:
\[ r_1 = \frac{D_1}{2},\quad r_2 = \frac{D_2}{2} \]
Then plug \(r_1\) and \(r_2\) into the standard formulas. A quick shortcut for volume is:
\[ V = \frac{\pi h}{12}\left(D_1^2 + D_1 D_2 + D_2^2\right) \]
What happens if the two radii are equal?
If \(r_1 = r_2 = r\), the frustum becomes a cylinder. The volume formula reduces to:
\[ V = \frac{1}{3}\pi h (r^2 + r^2 + r^2) = \pi r^2 h \]
which is exactly the usual cylinder volume formula.
Do I need the slant height to compute the volume?
No. The volume depends only on the two radii and the vertical height. The slant height is needed for the lateral surface area, not for the volume.