Question

Surface Area of Pyramids Quick Check
2 of 52 of 5 Items
Question
Use the formula for the surface area of a square pyramid to find the surface area if the base edge length is 5 yards and the height is 25 yards. Round to the nearest whole number.(1 point)
Responses

251 square yards
251 square yards

1,298 square yards
1,298 square yards

276 square yards
276 square yards

151 square yards

Answer 1

To find the surface area of a square pyramid, you can use the formula:

\[
\text{Surface Area} = B + \frac{1}{2} \cdot P \cdot l
\]

where:
- \( B \) is the area of the base,
- \( P \) is the perimeter of the base,
- \( l \) is the slant height.

1. **Calculate the area of the base \( B \)**:
Since the base is a square with edge length of 5 yards:
\[
B = \text{edge length}^2 = 5^2 = 25 \text{ square yards}
\]

2. **Calculate the perimeter \( P \)**:
The perimeter of a square is given by:
\[
P = 4 \cdot \text{edge length} = 4 \cdot 5 = 20 \text{ yards}
\]

3. **Calculate the slant height \( l \)**:
The height of the pyramid is given as 25 yards. To find the slant height, you can use the Pythagorean theorem. The slant height \( l \) is the hypotenuse of a right triangle where one leg is half of the base edge length (which is \( \frac{5}{2} = 2.5 \) yards), and the other leg is the height of the pyramid (25 yards).

Thus:
\[
l = \sqrt{(\text{height})^2 + \left(\frac{\text{base edge length}}{2}\right)^2}
\]
\[
l = \sqrt{25^2 + 2.5^2} = \sqrt{625 + 6.25} = \sqrt{631.25} \approx 25.14 \text{ yards}
\]

4. **Now plug these values into the surface area formula**:
\[
\text{Surface Area} = B + \frac{1}{2} \cdot P \cdot l
\]
\[
\text{Surface Area} = 25 + \frac{1}{2} \cdot 20 \cdot 25.14
\]
\[
\text{Surface Area} = 25 + 10 \cdot 25.14 = 25 + 251.4 = 276.4
\]

5. **Round to the nearest whole number**:
The surface area is approximately 276.4, which rounds to 276 square yards.

So, the final answer is **276 square yards**.