An architect is designing a square-based pyramid for a public monument. The base has a perimeter of 60 m, and the total lateral surface area (the sum of the four triangular faces) is 500 m². To finalize the design, the architect needs to find the vertical height of the pyramid.

Solution

Solution: Base side \(a=60\div4=15\) m. Lateral surface area of a square pyramid is \(\frac12\times\text{perimeter}\times l\), where l is the slant height. Thus, \(500=\frac12\times60\times l\Rightarrow l=\frac{500}{30}=16.67\) m approximately. The vertical height is \(h=\sqrt{l^2-(a/2)^2}=\sqrt{16.67^2-7.5^2}\approx14.88\) m. Trick: Find the slant height from the lateral area first, then use Pythagoras to calculate the vertical height.