Question
surface area of a triangular prism with a
what is the surface area of a triangular prism with
width of 9
length of 6
height of 8
what is the surface area of a triangular prism with
width of 9
length of 6
height of 8
Answers
Answered by
GPT-4o mini
To find the surface area of a triangular prism, we need to determine the areas of the triangular bases and the three rectangular sides.
1. **Identify the dimensions of the triangular base**:
- For the triangular base given the width (base of the triangle) = 9 and height of the triangle = 8, we can calculate the area of the triangular base.
\[
\text{Area of the triangular base} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 8 = 36
\]
2. **Calculate the surface area contributions**:
The triangular prism has two identical triangular bases and three rectangular sides.
- The total area of the triangular bases:
\[
\text{Area of two triangular bases} = 2 \times 36 = 72
\]
3. **Determine the area of the rectangular sides**:
- The first side (base of the triangle):
\[
\text{Area} = \text{base} \times \text{length} = 9 \times 6 = 54
\]
- The second side (height of the triangle):
\[
\text{Area} = \text{height} \times \text{length} = 8 \times 6 = 48
\]
- The third side (the slant height of the triangle): To calculate this, we first need to determine the slant height. Assuming it's a right triangle, we can find it using the Pythagorean theorem:
\[
\text{slant height} = \sqrt{(\text{base}/2)^2 + \text{height}^2} = \sqrt{(9/2)^2 + 8^2} = \sqrt{20.25 + 64} = \sqrt{84.25} \approx 9.2
\]
Therefore,
\[
\text{Area} = \text{slant height} \times \text{length} \approx 9.2 \times 6 \approx 55.2
\]
4. **Add up all the areas**:
- Total surface area = Area of triangular bases + Area of rectangular sides
\[
\text{Total Surface Area} = 72 + 54 + 48 + 55.2 = 229.2
\]
So the total surface area of the triangular prism is approximately **229.2 square units**.
1. **Identify the dimensions of the triangular base**:
- For the triangular base given the width (base of the triangle) = 9 and height of the triangle = 8, we can calculate the area of the triangular base.
\[
\text{Area of the triangular base} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 8 = 36
\]
2. **Calculate the surface area contributions**:
The triangular prism has two identical triangular bases and three rectangular sides.
- The total area of the triangular bases:
\[
\text{Area of two triangular bases} = 2 \times 36 = 72
\]
3. **Determine the area of the rectangular sides**:
- The first side (base of the triangle):
\[
\text{Area} = \text{base} \times \text{length} = 9 \times 6 = 54
\]
- The second side (height of the triangle):
\[
\text{Area} = \text{height} \times \text{length} = 8 \times 6 = 48
\]
- The third side (the slant height of the triangle): To calculate this, we first need to determine the slant height. Assuming it's a right triangle, we can find it using the Pythagorean theorem:
\[
\text{slant height} = \sqrt{(\text{base}/2)^2 + \text{height}^2} = \sqrt{(9/2)^2 + 8^2} = \sqrt{20.25 + 64} = \sqrt{84.25} \approx 9.2
\]
Therefore,
\[
\text{Area} = \text{slant height} \times \text{length} \approx 9.2 \times 6 \approx 55.2
\]
4. **Add up all the areas**:
- Total surface area = Area of triangular bases + Area of rectangular sides
\[
\text{Total Surface Area} = 72 + 54 + 48 + 55.2 = 229.2
\]
So the total surface area of the triangular prism is approximately **229.2 square units**.