Question

find the surface area of a square pyramid with side length 3 in and slant height 5 in.
answer

Explanation:

Step1: Calculate the area of the base

The base is a square with side length \(s = 3\) in. The area of a square is \(A_{base}=s^{2}\). So, \(A_{base}=3^{2}=9\) \(in^{2}\).

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 (\(b = 3\) in) and the height is the slant height (\(l = 5\) in). So, \(A_{\triangle}=\frac{1}{2}\times3\times5=\frac{15}{2}=7.5\) \(in^{2}\).

Step3: Calculate the total area of the four triangular faces

Since there are 4 triangular faces, \(A_{triangles}=4\times A_{\triangle}\). Substituting the value of \(A_{\triangle}\), we get \(A_{triangles}=4\times7.5 = 30\) \(in^{2}\).

Step4: Calculate the total surface area of the square pyramid

The total surface area \(A\) of a square pyramid is the sum of the area of the base and the area of the four triangular faces. So, \(A=A_{base}+A_{triangles}\). Substituting the values of \(A_{base}\) and \(A_{triangles}\), we get \(A = 9+30=39\) \(in^{2}\).

Answer:

\(39\)