Discussion
A square pyramid has a base perimeter x, and the slant height is half of the perimeter. What is the lateral surface area of the pyramid?
Answer: D
The lateral surface area is \(0.25\,x^{2}\).
The base is a square of perimeter \(x\), so each side is \(x/4\). The slant height is half the perimeter, \(x/2\). The lateral surface consists of four congruent triangles, each with base \(x/4\) and height equal to the slant height:
\[ \begin{aligned} A &= 4 \times \frac{1}{2} \times \frac{x}{4} \times \frac{x}{2} \\ &= \frac{x^{2}}{4} = 0.25\,x^{2} \end{aligned} \]
\(0.50\,x^{2}\) forgets the half in the triangle area, and \(x^{2}\) uses the full perimeter as the base of each face. \(0.75\,x^{2}\) matches no consistent working.
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