Conical Frustum Volume Calculator
Find the volume of a truncated cone (frustum) — a bucket, lampshade or drinking cup — from its two radii and height, with slant height and surface area.
How it works
Volume = ⅓ · π · h · (R² + R·r + r²) · Slant ℓ = √(h² + (R − r)²) · Lateral = π · (R + r) · ℓ · Total SA = lateral + πR² + πr²
A conical frustum is what remains when you cut the top off a cone with a plane parallel to its base — the shape of a bucket, a plant pot or a paper coffee cup. Its volume interpolates between the two circular ends through the term R² + R·r + r², which reduces to the plain cone formula when the top radius r is zero. The slant height follows the Pythagorean theorem on the height and the difference in radii, and the curved side has area π(R + r)ℓ.
Worked example
For a bottom radius of 6, top radius of 3 and height 8: volume = ⅓ × π × 8 × (36 + 18 + 9) = ⅓ × π × 8 × 63 = 168π ≈ 527.7876 cubic units. The slant height is √(8² + (6 − 3)²) = √73 ≈ 8.5440, so the lateral area is π × 9 × 8.5440 ≈ 241.5468 and the total surface area adds both caps, πR² + πr² = 45π ≈ 141.3717.
Sources: Frustum — Wikipedia
Frequently asked questions
What is a frustum?
A frustum is the portion of a cone or pyramid lying between two parallel planes — the solid left after slicing off the pointed top. Everyday frustums include lampshades, buckets and the classic paper cup.
How does this relate to the plain cone formula?
Set the top radius r to zero and R² + R·r + r² collapses to R², giving ⅓πR²h — the ordinary cone volume. The frustum formula is the general case that contains the cone as a special value.
Why is the slant height not just the height?
The sides slope outward, so the distance measured along the surface is longer than the vertical height. It equals √(h² + (R − r)²): the vertical rise combined with the horizontal change in radius, via the Pythagorean theorem.
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This calculator is for educational purposes. Double-check important results before acting on them.