Question
Show your steps and explain how to find the surface area of the triangular prism below.
6 in
8 in
10 in
19 in
The base of the triangle (b) = 8 inches
The height of the triangle (h) = 6 inches
The length of the prism (L) = 19 inches
(4 points)
6 in
8 in
10 in
19 in
The base of the triangle (b) = 8 inches
The height of the triangle (h) = 6 inches
The length of the prism (L) = 19 inches
(4 points)
Answers
Answered by
GPT-4o mini
To find the surface area of a triangular prism, you need to calculate the area of the triangular bases and the area of the three rectangular sides (lateral surfaces) of the prism. Here’s how you can do it step-by-step:
### Step 1: Calculate the Area of the Triangular Base
The formula for the area of a triangle is:
\[
\text{Area of triangle} = \frac{1}{2} \times b \times h
\]
where:
- \( b \) is the base of the triangle
- \( h \) is the height of the triangle
Given:
- \( b = 8 \) inches
- \( h = 6 \) inches
Now plug in the values:
\[
\text{Area of triangle} = \frac{1}{2} \times 8 \times 6 = \frac{1}{2} \times 48 = 24 \text{ square inches}
\]
### Step 2: Calculate the Area of the Rectangles
The prism has three rectangular faces. Each rectangle can be calculated based on its dimensions:
1. **Rectangle 1 (base of the triangle)**:
- This rectangle has a width equal to the base of the triangular base, and a height equal to the length of the prism.
\[
\text{Area} = b \times L = 8 \times 19 = 152 \text{ square inches}
\]
2. **Rectangle 2 (side of the triangle)**:
- This rectangle has a height equal to the length of the prism and a width equal to one side of the triangle (the length of one of the triangle's sides, which is \( 10 \) inches based on the given dimensions).
\[
\text{Area} = \text{Side} \times L = 10 \times 19 = 190 \text{ square inches}
\]
3. **Rectangle 3 (other side of the triangle)**:
- To find the length of the third side of the triangle, we can use the Pythagorean theorem since we have a right triangle (with the base and height as the two legs). The third side (hypotenuse) can be calculated as follows:
\[
c = \sqrt{b^2 + h^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ inches}
\]
- This means Rectangle 3 uses the side of 10 inches as well.
\[
\text{Area} = \text{Other Side} \times L = 10 \times 19 = 190 \text{ square inches}
\]
### Step 3: Calculate the Total Surface Area
The total surface area (SA) is the sum of the areas of the two triangular bases and the three rectangular lateral faces:
1. **Area of two triangular bases**:
\[
\text{Total triangular area} = 2 \times \text{Area of triangle} = 2 \times 24 = 48 \text{ square inches}
\]
2. **Total area of the rectangular sides**:
\[
\text{Total rectangular area} = \text{Area of Rectangle 1} + \text{Area of Rectangle 2} + \text{Area of Rectangle 3} = 152 + 190 + 190 = 532 \text{ square inches}
\]
3. **Combine**:
\[
\text{Total Surface Area (SA)} = \text{Total triangular area} + \text{Total rectangular area} = 48 + 532 = 580 \text{ square inches}
\]
### Final Answer
Thus, the total surface area of the triangular prism is:
\[
\text{Surface Area} = 580 \text{ square inches}
\]
### Step 1: Calculate the Area of the Triangular Base
The formula for the area of a triangle is:
\[
\text{Area of triangle} = \frac{1}{2} \times b \times h
\]
where:
- \( b \) is the base of the triangle
- \( h \) is the height of the triangle
Given:
- \( b = 8 \) inches
- \( h = 6 \) inches
Now plug in the values:
\[
\text{Area of triangle} = \frac{1}{2} \times 8 \times 6 = \frac{1}{2} \times 48 = 24 \text{ square inches}
\]
### Step 2: Calculate the Area of the Rectangles
The prism has three rectangular faces. Each rectangle can be calculated based on its dimensions:
1. **Rectangle 1 (base of the triangle)**:
- This rectangle has a width equal to the base of the triangular base, and a height equal to the length of the prism.
\[
\text{Area} = b \times L = 8 \times 19 = 152 \text{ square inches}
\]
2. **Rectangle 2 (side of the triangle)**:
- This rectangle has a height equal to the length of the prism and a width equal to one side of the triangle (the length of one of the triangle's sides, which is \( 10 \) inches based on the given dimensions).
\[
\text{Area} = \text{Side} \times L = 10 \times 19 = 190 \text{ square inches}
\]
3. **Rectangle 3 (other side of the triangle)**:
- To find the length of the third side of the triangle, we can use the Pythagorean theorem since we have a right triangle (with the base and height as the two legs). The third side (hypotenuse) can be calculated as follows:
\[
c = \sqrt{b^2 + h^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ inches}
\]
- This means Rectangle 3 uses the side of 10 inches as well.
\[
\text{Area} = \text{Other Side} \times L = 10 \times 19 = 190 \text{ square inches}
\]
### Step 3: Calculate the Total Surface Area
The total surface area (SA) is the sum of the areas of the two triangular bases and the three rectangular lateral faces:
1. **Area of two triangular bases**:
\[
\text{Total triangular area} = 2 \times \text{Area of triangle} = 2 \times 24 = 48 \text{ square inches}
\]
2. **Total area of the rectangular sides**:
\[
\text{Total rectangular area} = \text{Area of Rectangle 1} + \text{Area of Rectangle 2} + \text{Area of Rectangle 3} = 152 + 190 + 190 = 532 \text{ square inches}
\]
3. **Combine**:
\[
\text{Total Surface Area (SA)} = \text{Total triangular area} + \text{Total rectangular area} = 48 + 532 = 580 \text{ square inches}
\]
### Final Answer
Thus, the total surface area of the triangular prism is:
\[
\text{Surface Area} = 580 \text{ square inches}
\]