Geometry · Frustum volume derivation
A circular frustum uses the volume formula V = (1/3)πh(R² + Rr + r²) because it is the part of a large cone left after a smaller, similar cone is removed. Similarity converts the two cone heights into R, r, and h, while disk integration confirms the same three-term expression.
The R² and r² terms represent the two circular ends. The middle term Rr accounts for the continuous change between them. It is not an average-radius shortcut: subtracting two similar cones or integrating the changing cross-sectional area produces exactly R² + Rr + r².
What Does The Frustum Volume Formula Describe?
A circular frustum is the solid left when a plane parallel to a cone’s base cuts off its tip. If the larger radius is R, the smaller radius is r, and the perpendicular height is h, its volume is V = (1/3)πh(R² + Rr + r²).
The two circular faces must be parallel and centered on the same axis. The height h is the shortest distance between those faces, not the slanted side length. The radii and height use linear units, so the resulting volume uses cubic units.
The formula works even if the labels R and r are swapped because R² + Rr + r² is symmetric. Calling one radius “large” and the other “small” helps with the cone-subtraction proof, but the final expression does not depend on that naming choice.
This article explains why the formula works. To calculate a result directly from dimensions, use the frustum calculator. The derivation below focuses on the geometry behind the tool rather than repeating its input instructions.
How Does A Frustum Come From Two Similar Cones?
Extend the sloping sides of the frustum until they meet at an imaginary tip. The completed shape is a large cone, and the missing top is a smaller cone. Because the cutting plane is parallel to the base, the two cones have the same shape and their radii are proportional to their heights.
Let H be the height of the completed large cone. The removed small cone then has height H − h. Similarity gives R/H = r/(H − h). Cross-multiplying produces R(H − h) = rH, so H(R − r) = Rh.
When R is greater than r, the completed height is H = Rh/(R − r). The small cone’s height is H − h = rh/(R − r). These expressions are not extra measurements; they translate the hidden cone heights into the three dimensions already known.
The similarity step is the heart of the subtraction proof. It ensures the imaginary extension has the same slope as the actual frustum. Brown University’s geometric discussion of conical frustums likewise treats the solid through corresponding sections of a cone (Brown University, accessed October 14, 2026).
How Does Cone Subtraction Produce The Formula?
A cone with radius a and height b has volume (1/3)πa²b. Therefore, the frustum volume equals the large cone volume minus the small cone volume:
V = (1/3)πR²H − (1/3)πr²(H − h).
Substitute H = Rh/(R − r) and H − h = rh/(R − r):
V = (πh/3) × (R³ − r³)/(R − r).
The difference-of-cubes identity says R³ − r³ = (R − r)(R² + Rr + r²). Canceling R − r gives the final result:
V = (1/3)πh(R² + Rr + r²).
This cancellation also explains why the final formula remains valid when R = r, even though the intermediate height H would involve division by zero. The completed cone moves infinitely far away in that limiting picture, but the simplified formula becomes the ordinary cylinder formula.
The method uses the same cone volume rule explained by the cone calculator. Its new contribution is the similarity relationship that removes the imaginary heights from the answer.
Why Does The Middle Rr Term Appear?
The term Rr appears because a frustum’s radius changes continuously, so its volume depends on the interaction between both end radii. Using only R² and r² would miss the intermediate circular slices. Algebra records that interaction when the difference of two cubes is divided by the difference of their radii.
It may be tempting to average the radii first and treat the solid like a cylinder. That estimate would be πh[(R + r)/2]², which expands to (πh/4)(R² + 2Rr + r²). Its coefficients differ from the exact one-third formula, so it generally gives the wrong volume.
The cross-term is therefore not a correction chosen to fit examples. It appears independently in the cone subtraction and in the integral of the changing disk area. Two different methods producing the same term is a strong consistency check.
How Does Disk Integration Confirm The Formula?
Place x = 0 at the face of radius r and x = h at the face of radius R. Because the side is straight, the radius changes linearly: ρ(x) = r + (R − r)x/h. A thin perpendicular slice is a disk with area A(x) = π[ρ(x)]².
The volume is the sum of those thin disks, expressed as the integral V = ∫₀ʰ π[r + (R − r)x/h]² dx. Calculus texts derive volumes of revolution by integrating cross-sectional disk areas in this way (University of Kentucky, accessed October 14, 2026).
Expand the square and integrate each part:
V = π∫₀ʰ [r² + 2r(R − r)x/h + (R − r)²x²/h²] dx.
After evaluation, V = πh[r² + r(R − r) + (R − r)²/3]. Collecting like terms gives (1/3)πh(R² + Rr + r²), exactly matching the similar-cone derivation. Ohio State University’s calculus materials use the same disk and washer principle for solids of revolution (Ohio State University, accessed October 14, 2026).
How Do You Check A Frustum Example?
Take R = 6, r = 3, and h = 4. The exact volume is V = (1/3)π(4)(6² + 6 × 3 + 3²). The expression in parentheses is 36 + 18 + 9 = 63, so V = 84π cubic units.
Using π ≈ 3.141592654 gives V ≈ 263.8938 cubic units. A useful size check is that this frustum must be smaller than a cylinder of radius 6 and height 4, whose volume is 144π, but larger than a cylinder of radius 3 and height 4, whose volume is 36π. The result 84π lies between them.
You can also reconstruct the cones. The large cone height is H = 6 × 4/(6 − 3) = 8, and the small cone height is 4. Their volumes are 96π and 12π, respectively. The difference is again 84π.
What Do The Cone And Cylinder Limits Prove?
Setting r = 0 turns the frustum into a cone, so the formula becomes V = (1/3)πhR². For R = 6 and h = 4, that is 48π, or approximately 150.7964 cubic units. This agrees with the ordinary cone formula.
Setting R = r turns the sloping side into a vertical side and produces a cylinder. The three terms become r² + r² + r² = 3r², so V = πhr². For r = 3 and h = 5, the result is 45π, or approximately 141.3717 cubic units.
These limits are more than special examples. They show the formula connects smoothly to two familiar solids. Readers can compare the second result with the cylinder calculator or explore related solids through the trigonometry and geometry hub.
What Conditions And Mistakes Matter Most?
The formula assumes two parallel circular faces, a straight conical side, nonnegative radii, and a perpendicular height h ≥ 0. If the cut is tilted, the end face is not the parallel circle used in this derivation. A different geometric treatment may be needed.
- Do not use slant height as h. Volume uses the perpendicular distance between the parallel faces.
- Do not omit the Rr term. It represents the changing cross-sections between the two ends.
- Square each radius before adding. The expression is R² + Rr + r², not (R + r)².
- Keep units consistent. Convert all dimensions to one unit before calculating, then report cubic units.
- Do not round early. Keep π and exact values until the final decimal step.
If h = 0, the formula gives zero volume, as it should for a solid with no thickness. If both radii are zero, the solid collapses to a line and again has zero volume. These boundary checks help detect impossible or incorrectly entered dimensions.