Pyramids and perpendicular height

A pyramid has a polygonal base and triangular faces meeting at an apex. Its volume is \(V=\dfrac13Ah\), where \(A\) is base area and \(h\) is perpendicular height from the apex to the base plane. A face’s slant height is the perpendicular altitude within that triangular face, not a sloping edge.

Example

A right square-based pyramid has base side 10 cm and perpendicular height 12 cm. Its volume is \(\dfrac13(10^2)(12)=400\text{ cm}^3\). The slant height of each triangular face is \(\sqrt{12^2+5^2}=13\text{ cm}\), using the distance from the base centre to a side midpoint.

Total surface area \(=10^2+4(\dfrac12\times10\times13)=360\text{ cm}^2\). The apex-to-corner edge is a different length, \(\sqrt{12^2+5^2+5^2}=\sqrt{194}\text{ cm}\), and cannot replace the face altitude in the triangle-area calculation.

For a general pyramid, add the areas of its individual faces; they need not share a common slant height. Omit the base only when it is not an exposed surface.