Question

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

Explanation:

Step1: Calculate base - area

The base is a square with side length $s = 3$ in. The area of the base $B=s^{2}=3^{2}=9$ in².

Step2: Calculate area of one triangular face

The formula for the area of a triangle is $A_{\triangle}=\frac{1}{2}bh$, where the base $b$ of the triangular face of the square - pyramid is the side - length of the base of the pyramid ($b = 3$ in) and the height is the slant height $l = 5$ in. So, $A_{\triangle}=\frac{1}{2}\times3\times5 = 7.5$ in².

Step3: Calculate total area of four triangular faces

Since there are 4 triangular faces, $A_{triangles}=4\times A_{\triangle}=4\times7.5 = 30$ in².

Step4: Calculate surface area

The surface area $SA$ of a square - pyramid is the sum of the base area and the total area of the triangular faces. So, $SA=B + A_{triangles}=9+30=39$ in².

Answer:

39