Question
Question
What is the surface area of this square pyramid?
10mm
12mm
(1 point)
Responses
624 sq. mm
624 sq. mm
144 sq. mm.
144 sq. mm.
240 sq. mm.
What is the surface area of this square pyramid?
10mm
12mm
(1 point)
Responses
624 sq. mm
624 sq. mm
144 sq. mm.
144 sq. mm.
240 sq. mm.
Answers
GPT-4o mini
To find the surface area of a square pyramid, we need to add the area of the square base to the area of the four triangular sides.
1. **Area of the base:** The base is a square with each side measuring 10 mm.
\[
\text{Area of the base} = \text{side}^2 = 10 \, \text{mm} \times 10 \, \text{mm} = 100 \, \text{sq. mm}
\]
2. **Area of the triangular sides:** Each triangular side has a base of 10 mm and a height that we need to calculate. The height of the triangular side can be found using the Pythagorean theorem since we know the height of the pyramid (12 mm) and half the base (5 mm).
Let's calculate the slant height (l):
\[
l = \sqrt{(\text{height}^2 + (\text{half the base})^2)} = \sqrt{(12^2 + 5^2)} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{mm}
\]
3. **Area of one triangular side:**
The area of one triangular side is given by:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{mm} \times 13 \, \text{mm} = 65 \, \text{sq. mm}
\]
4. **Area of four triangular sides:** Since there are four triangular sides:
\[
\text{Total area of the four triangles} = 4 \times 65 \, \text{sq. mm} = 260 \, \text{sq. mm}
\]
5. **Total surface area of the pyramid:**
\[
\text{Total surface area} = \text{Area of the base} + \text{Total area of the four triangles} = 100 \, \text{sq. mm} + 260 \, \text{sq. mm} = 360 \, \text{sq. mm}
\]
It seems none of the options provided (624 sq. mm, 144 sq. mm, or 240 sq. mm) are correct for this square pyramid based on the calculations—unless there was a misunderstanding regarding the dimensions. Please verify the provided dimensions or choices.
1. **Area of the base:** The base is a square with each side measuring 10 mm.
\[
\text{Area of the base} = \text{side}^2 = 10 \, \text{mm} \times 10 \, \text{mm} = 100 \, \text{sq. mm}
\]
2. **Area of the triangular sides:** Each triangular side has a base of 10 mm and a height that we need to calculate. The height of the triangular side can be found using the Pythagorean theorem since we know the height of the pyramid (12 mm) and half the base (5 mm).
Let's calculate the slant height (l):
\[
l = \sqrt{(\text{height}^2 + (\text{half the base})^2)} = \sqrt{(12^2 + 5^2)} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{mm}
\]
3. **Area of one triangular side:**
The area of one triangular side is given by:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{mm} \times 13 \, \text{mm} = 65 \, \text{sq. mm}
\]
4. **Area of four triangular sides:** Since there are four triangular sides:
\[
\text{Total area of the four triangles} = 4 \times 65 \, \text{sq. mm} = 260 \, \text{sq. mm}
\]
5. **Total surface area of the pyramid:**
\[
\text{Total surface area} = \text{Area of the base} + \text{Total area of the four triangles} = 100 \, \text{sq. mm} + 260 \, \text{sq. mm} = 360 \, \text{sq. mm}
\]
It seems none of the options provided (624 sq. mm, 144 sq. mm, or 240 sq. mm) are correct for this square pyramid based on the calculations—unless there was a misunderstanding regarding the dimensions. Please verify the provided dimensions or choices.