Question
calculate the surface area of a square pyramid that has a base length of 12 inches and a height of 18 inches. show your work by plugging in the dimensions of the pyramid into the correct surface area formula. explain each step of your solving process.
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GPT-4o mini
To calculate the surface area of a square pyramid, we need to consider both the area of the base and the area of the triangular faces. The formula for the surface area \( S \) of a square pyramid can be expressed as:
\[
S = B + L
\]
where \( B \) is the area of the base and \( L \) is the area of the lateral faces (the triangles).
### Step 1: Calculate the area of the base \( B \)
The base of the pyramid is a square. The formula for the area of a square is given by:
\[
B = \text{side}^2
\]
In this case, the side length of the base is 12 inches. Plugging in this value:
\[
B = 12^2 = 144 \, \text{square inches}
\]
### Step 2: Calculate the area of the lateral faces \( L \)
The surface area of the lateral faces is made up of 4 triangular faces. Each triangular face can be calculated using the area formula for a triangle:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of each triangle is the same as the side of the square base, which is 12 inches. The height of each triangular face is not the same as the height of the pyramid (18 inches). We need to calculate the slant height of the pyramid to use it as the height of the triangular sides of the pyramid.
### Step 3: Calculate the slant height \( l \)
To find the slant height, we can use the Pythagorean theorem. The height of the pyramid is the vertical height (\( h = 18 \) inches), and the distance from the center of the base to the midpoint of one side serves as one leg of the right triangle. Since the base is 12 inches, the distance from the center to the midpoint of a side is:
\[
\frac{12}{2} = 6 \, \text{inches}
\]
Now we have a right triangle where one leg is \( 6 \) inches (half the base length) and the other leg is \( 18 \) inches (the height of the pyramid).
Using the Pythagorean theorem:
\[
l = \sqrt{(\text{height})^2 + (\text{half base})^2} = \sqrt{18^2 + 6^2}
\]
\[
l = \sqrt{324 + 36} = \sqrt{360} = 6\sqrt{10} \, \text{inches} \approx 18.97 \, \text{inches}
\]
### Step 4: Calculate the area of the lateral faces \( L \)
Now, the area of one triangular face is:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 12 \times l = \frac{1}{2} \times 12 \times 6\sqrt{10}
\]
Calculating this gives:
\[
\text{Area of one triangle} = 6 \times 6\sqrt{10} = 36\sqrt{10} \, \text{square inches}
\]
Since there are 4 triangular faces, the total lateral area \( L \) is:
\[
L = 4 \times 36\sqrt{10} = 144\sqrt{10} \, \text{square inches}
\]
### Step 5: Combine the areas to find the total surface area \( S \)
Now we can calculate the total surface area of the pyramid:
\[
S = B + L = 144 + 144\sqrt{10}
\]
\[
S \approx 144 + 144 \times 3.162 \approx 144 + 456.00 \approx 600 \, \text{square inches}
\]
Thus, the total surface area of the square pyramid is:
\[
S \approx 600 \, \text{square inches}
\]
\[
S = B + L
\]
where \( B \) is the area of the base and \( L \) is the area of the lateral faces (the triangles).
### Step 1: Calculate the area of the base \( B \)
The base of the pyramid is a square. The formula for the area of a square is given by:
\[
B = \text{side}^2
\]
In this case, the side length of the base is 12 inches. Plugging in this value:
\[
B = 12^2 = 144 \, \text{square inches}
\]
### Step 2: Calculate the area of the lateral faces \( L \)
The surface area of the lateral faces is made up of 4 triangular faces. Each triangular face can be calculated using the area formula for a triangle:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of each triangle is the same as the side of the square base, which is 12 inches. The height of each triangular face is not the same as the height of the pyramid (18 inches). We need to calculate the slant height of the pyramid to use it as the height of the triangular sides of the pyramid.
### Step 3: Calculate the slant height \( l \)
To find the slant height, we can use the Pythagorean theorem. The height of the pyramid is the vertical height (\( h = 18 \) inches), and the distance from the center of the base to the midpoint of one side serves as one leg of the right triangle. Since the base is 12 inches, the distance from the center to the midpoint of a side is:
\[
\frac{12}{2} = 6 \, \text{inches}
\]
Now we have a right triangle where one leg is \( 6 \) inches (half the base length) and the other leg is \( 18 \) inches (the height of the pyramid).
Using the Pythagorean theorem:
\[
l = \sqrt{(\text{height})^2 + (\text{half base})^2} = \sqrt{18^2 + 6^2}
\]
\[
l = \sqrt{324 + 36} = \sqrt{360} = 6\sqrt{10} \, \text{inches} \approx 18.97 \, \text{inches}
\]
### Step 4: Calculate the area of the lateral faces \( L \)
Now, the area of one triangular face is:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 12 \times l = \frac{1}{2} \times 12 \times 6\sqrt{10}
\]
Calculating this gives:
\[
\text{Area of one triangle} = 6 \times 6\sqrt{10} = 36\sqrt{10} \, \text{square inches}
\]
Since there are 4 triangular faces, the total lateral area \( L \) is:
\[
L = 4 \times 36\sqrt{10} = 144\sqrt{10} \, \text{square inches}
\]
### Step 5: Combine the areas to find the total surface area \( S \)
Now we can calculate the total surface area of the pyramid:
\[
S = B + L = 144 + 144\sqrt{10}
\]
\[
S \approx 144 + 144 \times 3.162 \approx 144 + 456.00 \approx 600 \, \text{square inches}
\]
Thus, the total surface area of the square pyramid is:
\[
S \approx 600 \, \text{square inches}
\]
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