Surface area of a frustum of a cone

The formula

• A = π(R² + r²) + π(R + r) · l

R and r are the radii of the bottom and top circles and l the slant height, the length of the sloping side measured along the surface. The first bracket is the two flat circles; the second term is the curved side, which for a cone is πRl and for a frustum has both radii.

A worked example

R = 5, r = 2 and l = 6.7 cm: the circles are π(25 + 4) = 29π and the side π · 7 · 6.7 = 46.9π, so A = 75.9π ≈ 238.4 cm². A lampshade with the same measurements, open at both ends, needs the side alone: 46.9π ≈ 147 cm² of fabric plus seams.

The slant from the height

A frustum is usually measured by its height h, not its slant. The two are joined by Pythagoras across the difference in radii: l = √(h² + (R − r)²). For h = 6 and radii 5 and 2, l = √(36 + 9) = √45 ≈ 6.7, which is where the example’s slant came from.

Where π(R + r)l comes from

Unrolled, the curved side is a band of an annulus. The cone’s lateral area is πRL with L the full slant; cutting off the top cone with slant L − l and radius r leaves πRL − πr(L − l), and similar triangles turn that into π(R + r)l. The side’s area is the mean circumference, π(R + r), times the slant.

Common mistakes

• Using the height in place of the slant. The side is longer than the frustum is tall, and the area comes out short.
• Counting both circles when one is open. A bucket has a bottom and no lid: πR² + π(R + r)l, or with the small circle if the bucket stands on its narrow end.
• Mixing it with the volume. πh(R² + Rr + r²) / 3 is the volume, with the height, not the slant.

Frequently asked questions