‘Find the surface area of a square pyramid with side length 5 mi and slant height = 4 mi. 5 mi 5 mI’
The Correct Answer and Explanation is:
To find the surface area of a square pyramid, we need to calculate the area of both the base and the triangular faces (lateral surface area).
Formula for the Surface Area of a Square Pyramid:
Surface Area=Base Area+Lateral Surface Area\text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area}Surface Area=Base Area+Lateral Surface Area
- Base Area:
Since the base of the pyramid is a square, the area of the base is given by: Base Area=side length2\text{Base Area} = \text{side length}^2Base Area=side length2 Given that the side length is 5 miles, we substitute: Base Area=52=25 mi2\text{Base Area} = 5^2 = 25 \, \text{mi}^2Base Area=52=25mi2 - Lateral Surface Area:
The lateral surface area consists of 4 triangular faces. The area of one triangle is: Area of one triangle=12×base of triangle×slant height\text{Area of one triangle} = \frac{1}{2} \times \text{base of triangle} \times \text{slant height}Area of one triangle=21×base of triangle×slant height The base of each triangle is the same as the side length of the square (5 miles), and the slant height is 4 miles. Therefore: Area of one triangle=12×5×4=10 mi2\text{Area of one triangle} = \frac{1}{2} \times 5 \times 4 = 10 \, \text{mi}^2Area of one triangle=21×5×4=10mi2 Since there are 4 triangular faces, the total lateral surface area is: Lateral Surface Area=4×10=40 mi2\text{Lateral Surface Area} = 4 \times 10 = 40 \, \text{mi}^2Lateral Surface Area=4×10=40mi2 - Total Surface Area:
Finally, the total surface area is the sum of the base area and the lateral surface area: Surface Area=25+40=65 mi2\text{Surface Area} = 25 + 40 = 65 \, \text{mi}^2Surface Area=25+40=65mi2
Final Answer:
The surface area of the square pyramid is 65 square miles.
