Asked by EvilKitty1125
Calculate the surface area of the square pyramid that has a base length of 12 and a slant height of 18. Show your work.
Make it look REALLY simple, school appropriate
Make it look REALLY simple, school appropriate
Answers
Answered by
GPT-4o mini
To calculate the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular faces.
1. **Area of the Base:**
The base of the pyramid is a square. The area of a square is calculated using the formula:
\[
\text{Area}_{\text{base}} = \text{side} \times \text{side}
\]
Given that the base length (side) is 12:
\[
\text{Area}_{\text{base}} = 12 \times 12 = 144 \text{ square units}
\]
2. **Area of the Triangular Faces:**
Each triangular face has a base equal to the side length of the square, which is 12, and a height equal to the slant height of the pyramid, which is 18. The area of one triangle is calculated using the formula:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For one triangular face:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 12 \times 18
\]
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 216 = 108 \text{ square units}
\]
3. **Total Area of the Four Triangular Faces:**
Since there are four triangular faces, we multiply the area of one triangle by 4:
\[
\text{Total Area}_{\text{triangles}} = 4 \times 108 = 432 \text{ square units}
\]
4. **Total Surface Area of the Pyramid:**
Finally, we add the area of the base to the total area of the triangular faces:
\[
\text{Total Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{triangles}}
\]
\[
\text{Total Surface Area} = 144 + 432 = 576 \text{ square units}
\]
Thus, the surface area of the square pyramid is:
\[
\boxed{576 \text{ square units}}
\]
1. **Area of the Base:**
The base of the pyramid is a square. The area of a square is calculated using the formula:
\[
\text{Area}_{\text{base}} = \text{side} \times \text{side}
\]
Given that the base length (side) is 12:
\[
\text{Area}_{\text{base}} = 12 \times 12 = 144 \text{ square units}
\]
2. **Area of the Triangular Faces:**
Each triangular face has a base equal to the side length of the square, which is 12, and a height equal to the slant height of the pyramid, which is 18. The area of one triangle is calculated using the formula:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For one triangular face:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 12 \times 18
\]
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times 216 = 108 \text{ square units}
\]
3. **Total Area of the Four Triangular Faces:**
Since there are four triangular faces, we multiply the area of one triangle by 4:
\[
\text{Total Area}_{\text{triangles}} = 4 \times 108 = 432 \text{ square units}
\]
4. **Total Surface Area of the Pyramid:**
Finally, we add the area of the base to the total area of the triangular faces:
\[
\text{Total Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{triangles}}
\]
\[
\text{Total Surface Area} = 144 + 432 = 576 \text{ square units}
\]
Thus, the surface area of the square pyramid is:
\[
\boxed{576 \text{ square units}}
\]
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