Truncated Cone Area Calculator

Surface area of a frustum of a cone

Slice a cone with a plane parallel to its base and the leftover piece is a conical frustum. Think of a flower-pot or a bucket. Call the larger radius R, the smaller one r and the height h. The slant height comes out to g = √((R − r)² + h²), the lateral area is A_l = π·(R + r)·g, and once you fold in both circular ends the total surface area becomes A_t = π·(R + r)·g + π·R² + π·r². Run the numbers for R = 10 cm, r = 5 cm, h = 12 cm and you get g = 13 cm with A_l = 195π ≈ 612.6 cm².

Applications

You reach for frustum surface area whenever you need to estimate paint on a conical cup or planter, work out the sheet metal for a cooling-tower cladding, cut the fabric for a boat sail assembled from frustum panels, size the paper or cloth for a lampshade, lay out packaging for conical containers, or handle surface conversions in CAD/DCM modelling when a tapered section gets unwrapped flat.

FAQ

Why does the lateral area use R + r and not the average? Unroll the side of the frustum and you get an annular sector with two arc lengths, 2πR and 2πr. Treat those as the parallel "widths" of a trapezoid with slant height g, and its flat area is (2πR + 2πr)/2 · g = π·(R + r)·g.

Is g the same as h? They aren't. h measures straight across, perpendicular to the two circular faces, while g runs at an angle along the side. The two only line up when R = r, which makes the shape a cylinder.

When do I include the two end discs? For open shapes like a lampshade or a sail, the lateral area on its own is enough. Add π·R² + π·r² only when both ends are sealed, as with closed packaging or a sealed tank.