Question

what is the surface area of the square pyramid shown below?
a. 625 ft²
b. 1,025 ft²
c. 1,600 ft²
d. 2,225 ft²

Explanation:

Step1: Calculate the base area

The base is a square with side length \(25\) ft. The area of a square is \(A = s^2\), so \(A_{base}=25\times25 = 625\) \(ft^2\).

Step2: Calculate the perimeter of the base

The perimeter of a square is \(P = 4s\), so \(P = 4\times25=100\) ft.

Step3: Calculate the lateral surface area

The formula for the lateral surface area of a pyramid is \(A_{lateral}=\frac{1}{2}Pl\), where \(l\) is the slant height. Here \(P = 100\) ft and \(l = 32\) ft. So \(A_{lateral}=\frac{1}{2}\times100\times32= 1600\) \(ft^2\). But wait, no, actually, the total surface area formula for a square - pyramid is \(SA=A_{base}+\frac{1}{2}Pl\).
Substitute \(A_{base}=625\), \(P = 100\), \(l = 32\) into the formula: \(SA=625+\frac{1}{2}\times100\times32\).
First calculate \(\frac{1}{2}\times100\times32 = 1600\), then \(SA=625 + 1600=2225\) \(ft^2\).

Answer:

D. \(2,225\) \(ft^2\)