Area of a Frustum of a Cone Calculator

The formula

V = ⅓πh(R² + Rr + r²)

Not the average of the two circles

The volume uses R² + Rr + r², not (R² + r²)/2. The extra Rr cross term matters: for R = 6 and r = 3 the correct factor is 63 while the naive average gives 22.5. A frustum holds more than intuition suggests because the wider end dominates.

Setting r = R gives a cylinder and r = 0 gives a cone, so this one formula covers all three.

Volume scales with the cube

Double every length and the surface area quadruples while the volume grows eightfold. This is why doubling a container’s dimensions gives eight times the capacity but only four times the material — and why scaling models up rarely works the way intuition suggests.

The one-third family

Cones and pyramids are always exactly one third of the prism or cylinder that would enclose them, whatever the base shape. A sphere sits at two thirds of its enclosing cylinder — the result Archimedes asked to have carved on his tomb.

Units

Enter every dimension in the same unit; volume comes out cubed and surface area squared. If you work in centimetres, one litre is 1,000 cm³, which is why the capacity row is shown on the box and cylinder pages.