Question

find the surface area of a square pyramid with side length 3 km and slant height 3 km.

Explanation:

Step1: Calculate the area of the base

The base is a square with side length \(s = 3\) km. The area of a square is \(A_{base}=s^{2}\).

$$A_{base}=3^{2}=9$$

Step2: Calculate the area of one triangular face

The formula for the area of a triangle is \(A_{\triangle}=\frac{1}{2}\times base\times height\). Here, the base of the triangular face is the side - length of the square base (\(b = 3\) km) and the height is the slant height (\(l = 3\) km). So, \(A_{\triangle}=\frac{1}{2}\times3\times3=\frac{9}{2}\)

Step3: Calculate the total area of the four triangular faces

Since there are 4 triangular faces, \(A_{triangles}=4\times A_{\triangle}\).

$$A_{triangles}=4\times\frac{9}{2}=18$$

Step4: Calculate the total surface area

The total surface area of a square pyramid \(A = A_{base}+A_{triangles}\)

$$A=9 + 18=27$$

Answer:

\(27\)