Question
16.6 cm 17.2 cm 20 cm 18cm
What is the surface of the rectangular pyramid?
1,643.2 cm*2
1,178.8 cm*2
102,787.3 cm*2
1,001.6 cm*2
What is the surface of the rectangular pyramid?
1,643.2 cm*2
1,178.8 cm*2
102,787.3 cm*2
1,001.6 cm*2
Answers
GPT-4o mini
To calculate the surface area of a rectangular pyramid, we need to first determine the base area and the area of the triangular faces.
Given the dimensions of the rectangular base:
- Length (l) = 20 cm
- Width (w) = 17.2 cm
- Height (h) of the pyramid (which we'll assume to be perpendicular to the base) needs to be calculated.
However, since you provided four dimensions, I'm interpreting that:
- Let's assume the base of the pyramid is 20 cm by 17.2 cm, and the height of the pyramid is 16.6 cm (since it seems to be the vertical height).
1. **Calculate the area of the rectangular base:**
\[
\text{Base Area} = l \times w = 20 \, \text{cm} \times 17.2 \, \text{cm} = 344 \, \text{cm}^2
\]
2. **Next, calculate the area of the triangular faces:**
We know that there are two triangles for the length (the longer sides) and two triangles for the width (the shorter sides).
**For triangular faces on the length (20 cm side):**
- The base of each triangle = width of the base = 17.2 cm
- The height (slant height) can be derived using Pythagorean theorem:
Let's find the slant height (s) from the midpoint of the base (10 cm) to the apex (with height h = 16.6 cm):
\[
s = \sqrt{(10)^2 + (16.6)^2} = \sqrt{100 + 275.56} = \sqrt{375.56} \approx 19.4 \, \text{cm}
\]
Area of one triangle (long side):
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 17.2 \, \text{cm} \times 19.4 \, \text{cm} \approx 166.44 \, \text{cm}^2
\]
Total area for the two triangular faces along the length:
\[
2 \times 166.44 \approx 332.88 \, \text{cm}^2
\]
**For triangular faces on the width (17.2 cm side):**
- The base of each triangle = length of the base = 20 cm
\[
s = \sqrt{(8.6)^2 + (16.6)^2} \approx \sqrt{73.96 + 275.56} \approx \sqrt{349.52} \approx 18.7 \, \text{cm}
\]
Area of one triangle (short side):
\[
\text{Area} = \frac{1}{2} \times 20 \, \text{cm} \times 18.7 \, \text{cm} \approx 187 \, \text{cm}^2
\]
Total area for the two triangular faces along the width:
\[
2 \times 187 \approx 374 \, \text{cm}^2
\]
3. **Total Surface Area:**
Finally, we sum up all areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area of triangular faces (length)} + \text{Area of triangular faces (width)}
\]
\[
= 344 + 332.88 + 374 \approx 1050.88 \, \text{cm}^2
\]
None of the provided options (1,643.2 cm², 1,178.8 cm², 102,787.3 cm², 1,001.6 cm²) exactly match this value. Thus, it seems there may be an error in the provided dimensions or some misunderstandings regarding the interpretation of these dimensions.
However, calculate those areas correctly as suggested and check for calculations if needed to find the closest matching possibility or clarify any missing details.
Given the dimensions of the rectangular base:
- Length (l) = 20 cm
- Width (w) = 17.2 cm
- Height (h) of the pyramid (which we'll assume to be perpendicular to the base) needs to be calculated.
However, since you provided four dimensions, I'm interpreting that:
- Let's assume the base of the pyramid is 20 cm by 17.2 cm, and the height of the pyramid is 16.6 cm (since it seems to be the vertical height).
1. **Calculate the area of the rectangular base:**
\[
\text{Base Area} = l \times w = 20 \, \text{cm} \times 17.2 \, \text{cm} = 344 \, \text{cm}^2
\]
2. **Next, calculate the area of the triangular faces:**
We know that there are two triangles for the length (the longer sides) and two triangles for the width (the shorter sides).
**For triangular faces on the length (20 cm side):**
- The base of each triangle = width of the base = 17.2 cm
- The height (slant height) can be derived using Pythagorean theorem:
Let's find the slant height (s) from the midpoint of the base (10 cm) to the apex (with height h = 16.6 cm):
\[
s = \sqrt{(10)^2 + (16.6)^2} = \sqrt{100 + 275.56} = \sqrt{375.56} \approx 19.4 \, \text{cm}
\]
Area of one triangle (long side):
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 17.2 \, \text{cm} \times 19.4 \, \text{cm} \approx 166.44 \, \text{cm}^2
\]
Total area for the two triangular faces along the length:
\[
2 \times 166.44 \approx 332.88 \, \text{cm}^2
\]
**For triangular faces on the width (17.2 cm side):**
- The base of each triangle = length of the base = 20 cm
\[
s = \sqrt{(8.6)^2 + (16.6)^2} \approx \sqrt{73.96 + 275.56} \approx \sqrt{349.52} \approx 18.7 \, \text{cm}
\]
Area of one triangle (short side):
\[
\text{Area} = \frac{1}{2} \times 20 \, \text{cm} \times 18.7 \, \text{cm} \approx 187 \, \text{cm}^2
\]
Total area for the two triangular faces along the width:
\[
2 \times 187 \approx 374 \, \text{cm}^2
\]
3. **Total Surface Area:**
Finally, we sum up all areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area of triangular faces (length)} + \text{Area of triangular faces (width)}
\]
\[
= 344 + 332.88 + 374 \approx 1050.88 \, \text{cm}^2
\]
None of the provided options (1,643.2 cm², 1,178.8 cm², 102,787.3 cm², 1,001.6 cm²) exactly match this value. Thus, it seems there may be an error in the provided dimensions or some misunderstandings regarding the interpretation of these dimensions.
However, calculate those areas correctly as suggested and check for calculations if needed to find the closest matching possibility or clarify any missing details.