Question
How do I calculate the surface area of a square pyramid with a base length of 10 inches and a height of 15 inches?
Answers
GPT-4o mini
To calculate the surface area of a square pyramid, you need to consider both the area of the square base and the areas of the four triangular faces.
1. **Calculate the area of the base:**
The base is a square, so its area (A_base) can be calculated using the formula:
\[
A_{\text{base}} = \text{side}^2
\]
For a base length of 10 inches:
\[
A_{\text{base}} = 10^2 = 100 \text{ square inches}
\]
2. **Calculate the area of the triangular faces:**
Each triangular face has a base equal to the side of the square base (10 inches). To find the height of each triangular face, you can use the Pythagorean theorem. The height of the pyramid (15 inches) and half the base (which is 5 inches) form a right triangle with the slant height of the triangular face.
First, find the slant height (l) using the Pythagorean theorem:
\[
l = \sqrt{\left(\frac{\text{base side}}{2}\right)^2 + \text{height}^2} = \sqrt{5^2 + 15^2} = \sqrt{25 + 225} = \sqrt{250} = 5\sqrt{10} \text{ inches}
\]
3. **Calculate the area of one triangular face:**
The area (A_triangle) of one triangular face is given by the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Using the base of 10 inches and the slant height we've calculated:
\[
A_{\text{triangle}} = \frac{1}{2} \times 10 \times 5\sqrt{10} = 25\sqrt{10} \text{ square inches}
\]
4. **Calculate the total area of the four triangular faces:**
Since there are four triangular faces:
\[
A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 25\sqrt{10} = 100\sqrt{10} \text{ square inches}
\]
5. **Calculate the total surface area:**
The total surface area (SA) of the pyramid is the sum of the area of the base and the area of the triangular faces:
\[
SA = A_{\text{base}} + A_{\text{triangles}} = 100 + 100\sqrt{10}
\]
Thus, the surface area of the square pyramid is:
\[
SA = 100 + 100\sqrt{10} \text{ square inches}
\]
1. **Calculate the area of the base:**
The base is a square, so its area (A_base) can be calculated using the formula:
\[
A_{\text{base}} = \text{side}^2
\]
For a base length of 10 inches:
\[
A_{\text{base}} = 10^2 = 100 \text{ square inches}
\]
2. **Calculate the area of the triangular faces:**
Each triangular face has a base equal to the side of the square base (10 inches). To find the height of each triangular face, you can use the Pythagorean theorem. The height of the pyramid (15 inches) and half the base (which is 5 inches) form a right triangle with the slant height of the triangular face.
First, find the slant height (l) using the Pythagorean theorem:
\[
l = \sqrt{\left(\frac{\text{base side}}{2}\right)^2 + \text{height}^2} = \sqrt{5^2 + 15^2} = \sqrt{25 + 225} = \sqrt{250} = 5\sqrt{10} \text{ inches}
\]
3. **Calculate the area of one triangular face:**
The area (A_triangle) of one triangular face is given by the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Using the base of 10 inches and the slant height we've calculated:
\[
A_{\text{triangle}} = \frac{1}{2} \times 10 \times 5\sqrt{10} = 25\sqrt{10} \text{ square inches}
\]
4. **Calculate the total area of the four triangular faces:**
Since there are four triangular faces:
\[
A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 25\sqrt{10} = 100\sqrt{10} \text{ square inches}
\]
5. **Calculate the total surface area:**
The total surface area (SA) of the pyramid is the sum of the area of the base and the area of the triangular faces:
\[
SA = A_{\text{base}} + A_{\text{triangles}} = 100 + 100\sqrt{10}
\]
Thus, the surface area of the square pyramid is:
\[
SA = 100 + 100\sqrt{10} \text{ square inches}
\]