Truncated Cone Volume Calculator — Frustum Volume Formula
Find the volume of a truncated cone (frustum) — a cone with its tip sliced off — by entering the bottom radius r₁, top radius r₂ and height h. The formula V = (π/3) × h × (r₁² + r₁r₂ + r₂²) works for any unit of length.
cm
cm
cm
Unit
V = (π/3) × h × (r₁² + r₁r₂ + r₂²)
410.5
cm³- 1
Radii bracket r₁² + r₁r₂ + r₂²
5² + 5 × 3 + 3² = 49 - 2
Scale factor π ÷ 3
1.047198The frustum volume formula uses π/3, matching the cone volume formula. - 3
Volume = (π ÷ 3) × h × bracket
1.047198 × 8 × 49 = 410.5014
How does this calculator work?
V = (π/3) × h × (r₁² + r₁r₂ + r₂²). Reduces to cone volume πr²h/3 when top radius = 0, and to cylinder volume πr²h when both radii are equal. Common frustum examples: buckets, cups, silos.
Formula
How this is calculated
A truncated cone (frustum) is formed when a plane parallel to the base cuts through a cone, removing the apex. The resulting solid has two parallel circular faces of radii r₁ and r₂ and a slanted lateral surface connecting them. The volume is derived by computing the volumes of two full cones — the original and the removed tip — and subtracting, which simplifies to V = (π/3) × h × (r₁² + r₁r₂ + r₂²).
When r₂ = 0 the formula gives V = πr₁²h / 3, the volume of a complete cone. When r₁ = r₂ = r it gives V = πr²h, the volume of a cylinder. For any values in between, the mixed term r₁r₂ accounts for the frustum geometry that neither a pure cone nor a cylinder captures.
The slant height s = √(h² + (r₁ − r₂)²) is also reported, as it is needed for surface-area calculations. Common real-world frustums include drinking cups, buckets, silos, wine glasses, lamp shades, and poured concrete footings.
Frequently asked questions
You can use the prismatoid rule (Simpson's rule): V = (h/6) × (A₁ + 4A_m + A₂), where A₁ = πr₁², A₂ = πr₂², and A_m = π((r₁+r₂)/2)² is the area at the mid-height. This gives the same result as the frustum formula and works for any prismatoid solid.
A cone has a single circular base and tapers to a point (apex). A frustum is the piece that remains after cutting off the top of a cone with a plane parallel to the base — it has two circular faces instead of one. When the top radius is zero, the frustum degenerates back into a full cone.
1 litre = 1000 cm³. So if your radii and height are in centimetres, divide the result by 1000 to get litres. To get US gallons, divide cm³ by 3785.41. If your measurements are in inches, multiply in³ by 0.016387 to get litres, or by 0.004329 to get US gallons.
Also known as
TG we-Calculate Editorial Team. (2026). Truncated Cone Volume Calculator — Frustum Volume Formula [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/truncated-cone-volume-calculator
TG we-Calculate Editorial Team. "Truncated Cone Volume Calculator — Frustum Volume Formula." TG we-Calculate. 2026. https://we-calculate.com/calculator/truncated-cone-volume-calculator.
TG we-Calculate Editorial Team, "Truncated Cone Volume Calculator — Frustum Volume Formula," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/truncated-cone-volume-calculator
@misc{wecalculate_truncated_cone_volume_calculator, title = {Truncated Cone Volume Calculator — Frustum Volume Formula}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/truncated-cone-volume-calculator}}, year = {2026}, note = {TG we-Calculate} }
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