Conical Frustum Calculator

Conical Frustum (Truncated Cone)

A conical frustum is the solid formed when a cone is cut by a plane parallel to its base.
The result has two circular faces, a smaller top circle and a larger bottom circle, joined by a curved lateral surface.

Formulas:

Variables:

• r1 = top radius (the smaller face)
• r2 = bottom radius (the larger base)
• h = perpendicular height between the two bases
• l = slant height measured along the curved side

Worked example: a tapered bucket, r1 = 8 cm, r2 = 12 cm, h = 15 cm

• Slant height: l = √(225 + 16) = √241 ≈ 15.52 cm
• Volume: (π × 15 / 3)(64 + 96 + 144) = 5π × 304 ≈ 4,775 cm³ (≈ 4.775 litres)
• Lateral SA: π(8 + 12)(15.52) ≈ 975 cm²
• Total SA: 975.4 + π(64) + π(144) ≈ 1,629 cm²

Special cases:

• If r1 = 0: the frustum reduces to a full cone
• If r1 = r2: the shape is a cylinder, and the formula still returns the right answer
• Swapping r1 and r2 gives the same volume (same shape, just flipped)

Real-world frustums:

Buckets, plastic drinking cups, flower pots, lamp shades and sheet-metal duct reducers all approximate a conical frustum. (Cooling towers, often listed here, are hyperboloids: their sides curve in and out again, which no frustum does.)
In engineering, the formula appears in volume calculations for tapered silos, liquid storage tanks with sloped walls, and structural columns that narrow toward the top.