Question
A block of wood has the shape of a triangular prism. The bases are right triangles. Find its surface area.
2 in.
2.5 in.
1.5 in.
17 in.
Question content area bottom
Part 1
The surface area is
enter your response here in.squared.
2 in.
2.5 in.
1.5 in.
17 in.
Question content area bottom
Part 1
The surface area is
enter your response here in.squared.
Answers
GPT-4o mini
To find the surface area of a triangular prism, we need to consider both the lateral area and the area of the triangular bases.
1. **Calculate the area of the triangular base:**
The base is a right triangle, and the area \(A\) of a right triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we can assume the dimensions provided (2 in, 2.5 in, and 1.5 in) refer to the legs of the right triangle. We first calculate the area of the triangular base.
Let's take:
- Base (leg 1) = 2 in
- Height (leg 2) = 2.5 in
Thus, the area \(A\) will be:
\[
A = \frac{1}{2} \times 2 \, \text{in} \times 2.5 \, \text{in} = \frac{1}{2} \times 5 \, \text{in}^2 = 2.5\, \text{in}^2
\]
Since there are two triangular bases, the total area of the triangular bases is:
\[
\text{Area of two bases} = 2 \times 2.5 \, \text{in}^2 = 5 \, \text{in}^2
\]
2. **Calculate the lateral area:**
The lateral area \(L\) of a prism is given by the perimeter of the base times the height (length of the prism). The perimeter \(P\) of a right triangle can be found using the lengths of the sides:
- The base is 2 in.
- The height is 2.5 in.
- The hypotenuse can be calculated using the Pythagorean theorem:
\[
c = \sqrt{a^2 + b^2} = \sqrt{(2\, \text{in})^2 + (2.5\, \text{in})^2} = \sqrt{4 + 6.25} = \sqrt{10.25} \approx 3.2\, \text{in}
\]
The perimeter \(P\) of the triangular base is:
\[
P = 2 \, \text{in} + 2.5 \, \text{in} + 3.2 \, \text{in} = 7.7 \, \text{in}
\]
Assuming the length of the prism (height of the prism) is 1.5 in, the lateral area \(L\) is:
\[
L = P \times \text{length} = 7.7 \, \text{in} \times 1.5 \, \text{in} = 11.55 \, \text{in}^2
\]
3. **Combine the areas to find the total surface area:**
The total surface area \(S\) of the triangular prism is the sum of the area of the bases and the lateral area:
\[
S = \text{Area of two bases} + \text{Lateral area} = 5 \, \text{in}^2 + 11.55 \, \text{in}^2 = 16.55 \, \text{in}^2
\]
Thus, the surface area of the triangular prism is approximately **16.55 square inches**.
### Summary
**Total Surface Area = 16.55 in²**
1. **Calculate the area of the triangular base:**
The base is a right triangle, and the area \(A\) of a right triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we can assume the dimensions provided (2 in, 2.5 in, and 1.5 in) refer to the legs of the right triangle. We first calculate the area of the triangular base.
Let's take:
- Base (leg 1) = 2 in
- Height (leg 2) = 2.5 in
Thus, the area \(A\) will be:
\[
A = \frac{1}{2} \times 2 \, \text{in} \times 2.5 \, \text{in} = \frac{1}{2} \times 5 \, \text{in}^2 = 2.5\, \text{in}^2
\]
Since there are two triangular bases, the total area of the triangular bases is:
\[
\text{Area of two bases} = 2 \times 2.5 \, \text{in}^2 = 5 \, \text{in}^2
\]
2. **Calculate the lateral area:**
The lateral area \(L\) of a prism is given by the perimeter of the base times the height (length of the prism). The perimeter \(P\) of a right triangle can be found using the lengths of the sides:
- The base is 2 in.
- The height is 2.5 in.
- The hypotenuse can be calculated using the Pythagorean theorem:
\[
c = \sqrt{a^2 + b^2} = \sqrt{(2\, \text{in})^2 + (2.5\, \text{in})^2} = \sqrt{4 + 6.25} = \sqrt{10.25} \approx 3.2\, \text{in}
\]
The perimeter \(P\) of the triangular base is:
\[
P = 2 \, \text{in} + 2.5 \, \text{in} + 3.2 \, \text{in} = 7.7 \, \text{in}
\]
Assuming the length of the prism (height of the prism) is 1.5 in, the lateral area \(L\) is:
\[
L = P \times \text{length} = 7.7 \, \text{in} \times 1.5 \, \text{in} = 11.55 \, \text{in}^2
\]
3. **Combine the areas to find the total surface area:**
The total surface area \(S\) of the triangular prism is the sum of the area of the bases and the lateral area:
\[
S = \text{Area of two bases} + \text{Lateral area} = 5 \, \text{in}^2 + 11.55 \, \text{in}^2 = 16.55 \, \text{in}^2
\]
Thus, the surface area of the triangular prism is approximately **16.55 square inches**.
### Summary
**Total Surface Area = 16.55 in²**