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
(4 points)
6 in
8 in
10 in
19 in
(4 points)
Answers
Answered by
GPT-4o mini
To find the surface area of a triangular prism, we need to calculate the area of the triangular base and the area of the three rectangular sides, then sum them up. A triangular prism has two triangular bases and three rectangular faces.
**Step 1: Identify the triangle dimensions and calculate the area of the triangular base.**
From the problem, we can assume:
- The base of the triangle (b) = 6 inches
- The height of the triangle (h) = 8 inches
- The length of the prism (L) = 19 inches
To find the area of the triangular base (A_triangle), we can use the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
\[
A_{\text{triangle}} = \frac{1}{2} \times 6 \, \text{in} \times 8 \, \text{in} = \frac{1}{2} \times 48 \, \text{in}^2 = 24 \, \text{in}^2
\]
Since the prism has two triangular bases, the total area of the triangular bases is:
\[
\text{Total area of triangular bases} = 2 \times A_{\text{triangle}} = 2 \times 24 \, \text{in}^2 = 48 \, \text{in}^2
\]
**Step 2: Calculate the area of the three rectangular faces.**
The three rectangular faces have the following dimensions:
1. Rectangle 1: Base side (b) = 6 inches, height = 19 inches
2. Rectangle 2: Height side (h) = 8 inches, height = 19 inches
3. Rectangle 3: Hypotenuse of the triangle (which we need to calculate)
First, we calculate the hypotenuse (c) of the triangle using the Pythagorean theorem:
\[
c = \sqrt{b^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{in}
\]
Now we calculate the areas of the rectangular faces:
- Area of Rectangle 1 (base side, 6 in):
\[
A_{\text{rectangle 1}} = \text{base} \times \text{length} = 6 \, \text{in} \times 19 \, \text{in} = 114 \, \text{in}^2
\]
- Area of Rectangle 2 (height side, 8 in):
\[
A_{\text{rectangle 2}} = \text{height} \times \text{length} = 8 \, \text{in} \times 19 \, \text{in} = 152 \, \text{in}^2
\]
- Area of Rectangle 3 (hypotenuse side, 10 in):
\[
A_{\text{rectangle 3}} = \text{hypotenuse} \times \text{length} = 10 \, \text{in} \times 19 \, \text{in} = 190 \, \text{in}^2
\]
**Step 3: Sum the area of all three rectangular faces:**
\[
\text{Total area of rectangular faces} = A_{\text{rectangle 1}} + A_{\text{rectangle 2}} + A_{\text{rectangle 3}}
\]
\[
\text{Total area of rectangular faces} = 114 \, \text{in}^2 + 152 \, \text{in}^2 + 190 \, \text{in}^2 = 456 \, \text{in}^2
\]
**Step 4: Calculate the total surface area of the prism:**
\[
\text{Total surface area} = \text{Total area of triangular bases} + \text{Total area of rectangular faces}
\]
\[
\text{Total surface area} = 48 \, \text{in}^2 + 456 \, \text{in}^2 = 504 \, \text{in}^2
\]
**Final Answer:**
The surface area of the triangular prism is **504 square inches**.
**Step 1: Identify the triangle dimensions and calculate the area of the triangular base.**
From the problem, we can assume:
- The base of the triangle (b) = 6 inches
- The height of the triangle (h) = 8 inches
- The length of the prism (L) = 19 inches
To find the area of the triangular base (A_triangle), we can use the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]
\[
A_{\text{triangle}} = \frac{1}{2} \times 6 \, \text{in} \times 8 \, \text{in} = \frac{1}{2} \times 48 \, \text{in}^2 = 24 \, \text{in}^2
\]
Since the prism has two triangular bases, the total area of the triangular bases is:
\[
\text{Total area of triangular bases} = 2 \times A_{\text{triangle}} = 2 \times 24 \, \text{in}^2 = 48 \, \text{in}^2
\]
**Step 2: Calculate the area of the three rectangular faces.**
The three rectangular faces have the following dimensions:
1. Rectangle 1: Base side (b) = 6 inches, height = 19 inches
2. Rectangle 2: Height side (h) = 8 inches, height = 19 inches
3. Rectangle 3: Hypotenuse of the triangle (which we need to calculate)
First, we calculate the hypotenuse (c) of the triangle using the Pythagorean theorem:
\[
c = \sqrt{b^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{in}
\]
Now we calculate the areas of the rectangular faces:
- Area of Rectangle 1 (base side, 6 in):
\[
A_{\text{rectangle 1}} = \text{base} \times \text{length} = 6 \, \text{in} \times 19 \, \text{in} = 114 \, \text{in}^2
\]
- Area of Rectangle 2 (height side, 8 in):
\[
A_{\text{rectangle 2}} = \text{height} \times \text{length} = 8 \, \text{in} \times 19 \, \text{in} = 152 \, \text{in}^2
\]
- Area of Rectangle 3 (hypotenuse side, 10 in):
\[
A_{\text{rectangle 3}} = \text{hypotenuse} \times \text{length} = 10 \, \text{in} \times 19 \, \text{in} = 190 \, \text{in}^2
\]
**Step 3: Sum the area of all three rectangular faces:**
\[
\text{Total area of rectangular faces} = A_{\text{rectangle 1}} + A_{\text{rectangle 2}} + A_{\text{rectangle 3}}
\]
\[
\text{Total area of rectangular faces} = 114 \, \text{in}^2 + 152 \, \text{in}^2 + 190 \, \text{in}^2 = 456 \, \text{in}^2
\]
**Step 4: Calculate the total surface area of the prism:**
\[
\text{Total surface area} = \text{Total area of triangular bases} + \text{Total area of rectangular faces}
\]
\[
\text{Total surface area} = 48 \, \text{in}^2 + 456 \, \text{in}^2 = 504 \, \text{in}^2
\]
**Final Answer:**
The surface area of the triangular prism is **504 square inches**.