Cone Frustum (Volume + Lateral)

Volume of a conical frustum

Slice a cone with a plane parallel to its base and the piece left behind is a conical frustum. Its volume comes out to V = (π·h/3)·(R² + r² + R·r). Here R is the larger radius at the bottom, r the smaller one at the top, and h the height measured straight up between the two faces.

Take R = 10 cm, r = 5 cm, h = 12 cm. That gives V = (π·12/3)·(100 + 25 + 50) = 4π·175 ≈ 2199 cm³. Two limits are worth noticing. Let r shrink to 0 and you get the full cone, V = π·R²·h/3; set R equal to r and the shape is just a cylinder, V = π·R²·h.

Applications

The shape turns up more often than you'd think: tapered drinking glasses, lampshades, buckets and pails, the disposable paper cones by a water dispenser. Power-plant cooling towers lean on it too, since their hyperbolic cross-section is close enough to a frustum for a quick volume estimate.

FAQ

Why three terms in the parentheses? The middle term R·r is the geometric mean of the two end areas. Drop it and the integral of the cross-sectional area along the axis no longer comes out right.

What if I have the slant height instead? Convert it first with h = √(s² − (R − r)²), where s is the slant.

Can I compute the lateral surface? You can. It's A = π·(R + r)·s, with s again the slant height.

How accurate is π = 3.14? On everyday objects the error sits around 0.05%. If your tolerances are tighter than that, switch to π ≈ 3.14159.