Cones, parallel cuts and frustums

For a right circular cone, \(V=\dfrac13\pi r^2h\), curved surface area \(=\pi r\ell\), and \(\ell^2=r^2+h^2\). The perpendicular height \(h\) and slant height \(\ell\) are different lengths.

A plane parallel to the base cuts off a smaller similar cone. The remaining solid is a frustum. Compare corresponding radii to find the removed cone’s height, then subtract its volume from the complete cone.

Frustum height \(=H-h\), not \(h\). For surface area, subtract the two curved areas and add only the circular faces actually exposed. An open top has no top disk; a closed solid has both circular faces.