What is the surface area of a square-based pyramid with a base edge length of 5 in. and a slant height of 8 in.?
A. 105 in ²
B. 145 in.2
C. 185 in 2
D. 224 in 2
The Correct Answer and Explanation is:
To find the surface area of a square-based pyramid, we use the following formula:
[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
]
Step 1: Calculate the Base Area
The base of the pyramid is a square. The area of a square is given by the formula:
[
\text{Base Area} = \text{side}^2
]
In this case, the side length of the square base is 5 inches, so:
[
\text{Base Area} = 5^2 = 25 \, \text{in}^2
]
Step 2: Calculate the Lateral Area
The lateral area of a square-based pyramid is the sum of the areas of the triangular faces. To calculate the area of one triangular face, we use the formula for the area of a triangle:
[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height}
]
The base of each triangular face is the side length of the square, which is 5 inches, and the slant height of the pyramid is given as 8 inches. So:
[
\text{Area of one triangle} = \frac{1}{2} \times 5 \times 8 = 20 \, \text{in}^2
]
Since the pyramid has four triangular faces, we multiply the area of one triangle by 4 to get the total lateral area:
[
\text{Lateral Area} = 4 \times 20 = 80 \, \text{in}^2
]
Step 3: Calculate the Total Surface Area
Now, we can add the base area and the lateral area to find the total surface area of the pyramid:
[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 25 + 80 = 105 \, \text{in}^2
]
Final Answer:
The surface area of the square-based pyramid is 105 in², so the correct answer is A. 105 in².
Explanation:
The formula for the surface area of a pyramid combines both the area of the square base and the total area of all the triangular faces. The slant height is used in the calculation of the area of the triangles, ensuring that we account for the slanted sides rather than just the vertical height of the pyramid. In this case, using the given base edge length and slant height leads to a surface area of 105 square inches.