Question

Find the surface area of a square pyramid with side length 5 mi and slant height 5 mi.
5 mi
5 mi
5 mi
Answer
Attempt 1 out of 2

Answer 1

To find the surface area of a square pyramid, we need to consider both the base area and the area of the triangular faces.

1. **Base Area**: Since the base is a square and the side length is 5 mi:
\[
\text{Base Area} = \text{side length}^2 = 5 \, \text{mi} \times 5 \, \text{mi} = 25 \, \text{mi}^2.
\]

2. **Triangular Faces Area**: A square pyramid has 4 triangular faces. The area of one triangular face can be calculated using the formula for the area of a triangle:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}.
\]
For the triangular face, the base is equal to the side length of the pyramid (5 mi), and the height is the slant height (5 mi):
\[
\text{Area of one triangle} = \frac{1}{2} \times 5 \, \text{mi} \times 5 \, \text{mi} = \frac{25}{2} \, \text{mi}^2 = 12.5 \, \text{mi}^2.
\]

Since there are 4 identical triangular faces:
\[
\text{Total area of triangular faces} = 4 \times 12.5 \, \text{mi}^2 = 50 \, \text{mi}^2.
\]

3. **Total Surface Area**: Now, we add the area of the base and the area of the triangular faces:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Total area of triangular faces} = 25 \, \text{mi}^2 + 50 \, \text{mi}^2 = 75 \, \text{mi}^2.
\]

Thus, the total surface area of the square pyramid is \( \boxed{75} \, \text{mi}^2 \).