The diagonal of the square base of a right square pyramid is 10√2 cm. If the lateral surface area of the pyramid is 260 cm², calculate the volume of the pyramid.

Solution

Solution: If the base side is a, then \(a\sqrt2=10\sqrt2\), so \(a=10\) cm. The lateral surface area of a square pyramid is \(2al\), where l is the slant height. Thus, \(2\times10\times l=260\Rightarrow l=13\) cm. The vertical height is \(h=\sqrt{13^2-(10/2)^2}=\sqrt{144}=12\) cm. Volume \(=\frac13\times10^2\times12=400\text{ cm}^3\). Trick: Find the base side from the diagonal, the slant height from the lateral area, and then the vertical height using Pythagoras.