Truncated Cone Surface Area Calculator (Frustum)

A truncated cone (conical frustum) has three surfaces: two parallel circular ends and the slanted lateral surface joining them.

This calculator returns the closed total, both ends plus the side. Most real frusta are open at one or both ends, so check the breakdown in the result rather than the headline. A bucket has one end, a lampshade has none.

SA = π × R² + π × r² + π × (R + r) × l

Where R is the larger radius, r is the smaller radius, h is the vertical height between the two circles, and l is the slant height:

l = √((R − r)² + h²)

The l accounts for the taper, and it is longer than h whenever R ≠ r.

Worked example: lampshade fabric

An empire shade tapers from 14 in across at the bottom to 10 in across at the top, standing 12 in tall. (A drum shade is the straight-sided kind, a cylinder; if yours has no taper, use the cylinder formula.)
R = 7 in, r = 5 in, h = 12 in.
Slant: l = √(4 + 144) = √148 ≈ 12.17 in.
Lateral surface, which is all the fabric there is: π × (7 + 5) × 12.17 ≈ 458.6 sq in = 3.18 sq ft.

But you cannot cut that from a 458 sq in rectangle. Flatten a frustum and it does not unroll to a rectangle, it unrolls to a curved band, part of an annulus. Extending the sides to the apex gives an outer radius of l × R / (R − r) = 42.6 in and an inner radius of 30.4 in, sweeping 59°. That crescent needs a rectangle roughly 42 in by 16 in to cut from, about 680 sq in of fabric for a 459 sq in shade, so close to a third of what you buy ends up as offcut. Lay the pattern out before ordering, and leave half an inch along one straight edge for the seam.

Worked example: galvanised steel bucket

A pail tapering from R = 7 in at the top to r = 5 in at the bottom, 11 in tall.
l = √(4 + 121) = √125 ≈ 11.18 in.
Lateral surface: π × 12 × 11.18 ≈ 421.4 sq in.
Bottom disc: π × 25 ≈ 78.5 sq in.
An open pail has no top, so the sheet metal comes to 500 sq in, about 3.47 sq ft, before seam allowance, handle and hardware.

Note that the calculator’s headline would say 654 sq in for those inputs, because it adds the top circle too. The open-pail figure is the one in the breakdown.

Where truncated cone surface matters in practice:

• Lampshades. Fabric, paper, parchment for shade materials.
• Buckets and pails. Sheet metal for galvanized or stainless steel buckets.
• Plastic pots. Tapered planter pots, where the surface area drives the injection-moulding shot weight.
• Hopper tank exteriors. Conical hoppers on industrial silos.
• Champagne flutes. Glass material for tapered drinking vessels.
• Conical heat shields. Aerospace re-entry vehicles use this shape.
• Beverage cups. Coffee shop hot cups taper from top rim to base.

The slant height formula details:

The slant l is NOT the same as h. For a perfectly cylindrical shape (R = r), l = h. For a truncated cone (R > r), the slant is longer than h because it includes the radial drop (R − r) and the vertical drop (h).

Visualise it with a string laid on the outside from rim to rim. The string travels (R − r) inward and h downward, and Pythagoras turns those two into the slant.

Open vs. closed truncated cones:

• Closed (both ends): total π(R² + r²) + π(R+r)l.
• Open at the wide end (bucket, plant pot): π × r² + π(R+r)l (the narrow end as a floor, plus the slant).
• Open both ends (lampshade, duct reducer): just π(R+r)l.

Pick the variant that matches the job. A bucket or a pot is open at the top, a lampshade or a duct transition is open at both ends, and a solid part like a cork or a rubber foot is closed all round.

Sanity check:

• R = r (no taper): SA = 2πR² + 2πRh. Matches cylinder formula. ✓
• r = 0 (full cone): SA = πR² + πR × √(R² + h²) = πR² + πR × slant. Matches cone formula. ✓