Cones and slant height

For a right circular cone, the perpendicular height \(h\), base radius \(r\) and slant height \(\ell\) form a right triangle: \(\ell^2=r^2+h^2\). Volume is \(\dfrac13\pi r^2h\); curved surface area is \(\pi r\ell\); closed total surface area is \(\pi r\ell+\pi r^2\).

Unrolling the curved surface gives a sector of radius \(\ell\), whose arc length equals the base circumference \(2\pi r\). Its area is \(\dfrac12\times\ell\times2\pi r=\pi r\ell\). The net’s sector radius is the slant height, not the cone’s base radius.

Example

A cone has radius 5 cm and height 12 cm, so \(\ell=13\text{ cm}\). Volume \(=100\pi\text{ cm}^3\), curved area \(=65\pi\text{ cm}^2\), and closed total area \(=90\pi\text{ cm}^2\). An open conical funnel with no circular base has only the curved area, ignoring thickness.

The volume formula uses perpendicular height; the curved-area formula uses slant height. These cannot be interchanged.