The volume of a frustum cone is V = (1/3)πh(R² + Rr + r²), where R is the radius of the larger circular base, r is the radius of the smaller circular top, and h is the perpendicular distance between the two bases. A frustum is the solid left behind when you slice a cone parallel to its base and discard the tip, which is why the formula looks like a slightly fancier version of the standard cone volume formula V = (1/3)πr²h — it accounts for two different end radii instead of one. Because it has two bases and a sloped side, a frustum shows up in real-world shapes like buckets, drinking cups, lampshades, and the conical hoppers used in industry, so being able to compute its volume is genuinely useful for sizing, capacity, and material estimates. The simplest way to get the answer is to plug R, r, and h into the formula above and multiply out; an alternative route, when the geometry works out cleanly, is to compute the full cone volume minus the missing tip cone volume, and a dedicated Cone Volume Calculator handles each of those cone volumes in a single step.
What a Frustum Is and How It Relates to a Full Cone
A frustum is the shape that remains after you cut a cone parallel to its circular base and remove the pointed tip. Picture an ice-cream cone with the tip snapped off, or a bucket that tapers from a wide opening at the top down to a smaller base — that tapered bucket is a frustum. The two flat ends are circles, with the larger one (radius R) at the wider end and the smaller one (radius r) at the narrower end. The slanted side connects them with straight lines.
Every frustum is part of a larger complete cone, and that relationship is the key to its volume formula. If you extended the slanted sides of a frustum upward until they met at a point, you would recover the original full cone. That is why the frustum formula V = (1/3)πh(R² + Rr + r²) is so closely related to the standard cone formula V = (1/3)πr²h — it is essentially the cone formula adjusted for a shape that has two different end radii instead of one.
The Frustum Volume Formula Explained
The volume of a frustum is V = (1/3)πh(R² + Rr + r²). Three measurements go in: the larger base radius R, the smaller top radius r, and the perpendicular height h between the two bases. The height h is measured straight up from one base to the other, parallel to the central axis of the shape — it is not the slant height along the slanted side.
Each part of the expression has a clear meaning. The factor (1/3)π is the same one that appears in every cone formula, because a cone of any kind fills exactly one-third of the cylinder that shares its base and height. The term h × (R² + Rr + r²) is what the integration of the cone's shrinking circular cross-sections collapses into. At the bottom, the cross-section is a circle of area πR²; at the top, it is a circle of area πr²; and the mixed term Rr is the average of the two end values, which is why it appears.
| Shape | Volume Formula | Limiting Case |
|---|---|---|
| Full cone | V = (1/3)πr²h | Frustum with r = 0 |
| Frustum | V = (1/3)πh(R² + Rr + r²) | Bridges the cone and cylinder |
| Cylinder | V = πr²h | Frustum with R = r |
When r = 0, the formula reduces to V = (1/3)πh(R² + R·0 + 0²) = (1/3)πR²h, which is just the regular cone formula. When R = r, the formula reduces to V = (1/3)πh(3R²) = πR²h, the cylinder formula. So the frustum formula gracefully bridges the cone and the cylinder.
How to Calculate Frustum Cone Volume Step by Step
Here is the direct, formula-based method, using R, r, and h:
- Measure or read the radius of the larger base, R.
- Measure or read the radius of the smaller base, r.
- Measure or read the perpendicular height, h, between the two bases (not the slant height along the side).
- Compute R², then R × r, then r².
- Add the three values together to get R² + Rr + r².
- Multiply that sum by h.
- Multiply by π.
- Divide by 3. The result is the volume in cubic units that match the unit you used for R, r, and h.
Here is the alternative subtraction method, which uses the full cone formula twice:
- Extend the slanted sides of the frustum upward until they meet at an apex; this reconstructs the original full cone.
- Using similar triangles, work out the radius R_big and the full height H_big of that original cone.
- Find the volume of the full cone with V = (1/3)π(R_big)² × H_big.
- Find the volume of the small "missing" tip cone with V = (1/3)π × r² × H_small, where H_small is the height from the top of the frustum up to the apex.
- Subtract the tip cone volume from the full cone volume. The result equals the frustum volume.
Using the Cone Volume Calculator for the Subtraction Method
If you go the subtraction route, you have to compute two cone volumes. The Cone Volume Calculator handles each one with the standard formula V = (1/3)πr²h and returns the value the moment you type the base radius and height. The calculator is unit-agnostic — enter the radius and height in centimeters and the result comes back in cubic centimeters (cm³), enter meters and you get cubic meters (m³), and so on. It also shows the base area, slant height, and lateral surface area alongside the volume, which makes it easy to cross-check the dimensions of each cone before subtracting.
To use it for the subtraction method, compute the full cone's radius and height once, type those two numbers into the calculator, and note the volume. Then compute the missing tip cone's radius and height, type those in, and note that volume too. The difference between the two is the frustum volume, which you can compare against the direct formula V = (1/3)πh(R² + Rr + r²) as a sanity check. The calculator runs locally in the browser, so the calculations are private and the answer is immediate.
A Worked Example
Suppose a conical bucket tapers from a top opening of radius r = 3 cm down to a base of radius R = 5 cm, with a perpendicular height of h = 6 cm. The frustum volume using the direct formula is:
V = (1/3)π × 6 × (5² + 5×3 + 3²) = (1/3)π × 6 × (25 + 15 + 9) = (1/3)π × 6 × 49 = 2π × 49 = 98π ≈ 307.88 cm³
To check via subtraction, extend the slanted sides until they meet. The slope from base to top is (R − r)/h = (5 − 3)/6 = 1/3, so the missing top cone (radius 3, sitting above the frustum) extends another 3 ÷ (1/3) = 9 cm above the frustum's top. That gives a tip cone of radius 3 and height 9, with volume (1/3)π × 9 × 9 = 27π ≈ 84.82 cm³. The full reconstructed cone has radius 5 and height 6 + 9 = 15, giving volume (1/3)π × 25 × 15 = 125π ≈ 392.70 cm³. Subtracting: 125π − 27π = 98π ≈ 307.88 cm³. The two methods agree, which is exactly what you want.
Where Frustum Volume Shows Up in Real Life
Frustum shapes appear in many practical objects, which is why being able to compute their volume is more than an exercise:
- Drinking cups and disposable paper cups are frustums, and a frustum volume calculation tells you how much liquid they actually hold.
- Plant pots, especially the classic tapered terracotta style, are frustums; the volume tells you how much soil they need.
- Storage hoppers and funnels are frustums at the bottom of silos, and engineers use the volume to plan material flow and inventory.
- Traffic cones, lampshades, and certain architectural columns are frustums; the volume determines mass, paint needed, or material cost.
- The frustum also appears in calculus and physics problems as the solid of revolution generated by rotating a line segment around an axis.
For any of these, the direct formula V = (1/3)πh(R² + Rr + r²) — or the subtraction method when you already have a full cone figure to work from — is the straightforward path to the answer.