Conical Frustum Calculator
Calculate volume and surface area of a conical frustum (truncated cone) from top radius, bottom radius, and height.
Includes slant height formula.
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 — connected by a curved lateral surface.
Formulas:
| Property | Formula |
|---|---|
| Slant Height | l = sqrt(h^2 + (r2 - r1)^2) |
| Volume | V = (pi * h / 3)(r1^2 + r1*r2 + r2^2) |
| Lateral Surface Area | LSA = pi(r1 + r2) * l |
| Top Circle Area | A1 = pi * r1^2 |
| Bottom Circle Area | A2 = pi * r2^2 |
| Total Surface Area | TSA = LSA + A1 + A2 |
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 — tapered bucket (r1 = 8 cm, r2 = 12 cm, h = 15 cm):
- Slant height: l = sqrt(225 + 16) = sqrt(241) ≈ 15.52 cm
- Volume: (pi × 15 / 3)(64 + 96 + 144) = 5pi × 304 ≈ 4,775 cm^3 (≈ 4.775 litres)
- Lateral SA: pi(8 + 12)(15.52) ≈ 975 cm^2
- Total SA: 975 + pi(64) + pi(144) ≈ 1,628 cm^2
Special cases:
- If r1 = 0: the frustum reduces to a full cone
- If r1 = r2: the shape is a cylinder — use the cylinder formula instead
- Swapping r1 and r2 gives the same volume (same shape, just flipped)
Real-world frustums: Buckets, plastic drinking cups, flower pots, lamp shades, and cooling tower sections all approximate a conical frustum. 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.
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