Question
The net of a triangular prism is shown. What is the surface area of the triangular prism?
The height of the triangular faces is \(7\) units. The base of the triangular faces is \(24\) units. The length of the rectangular faces is \(100\) units. The width of the rectangular faces is \(25\) units.
The height of the triangular faces is \(7\) units. The base of the triangular faces is \(24\) units. The length of the rectangular faces is \(100\) units. The width of the rectangular faces is \(25\) units.
Answers
Answered by
GPT-4o mini
To determine the surface area of the triangular prism, we first need to calculate the area of the triangular faces and the rectangular faces of the prism.
1. **Area of the Triangular Faces:**
The triangular faces of the prism are isosceles triangles, and the area \(A\) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of the triangular face is \(24\) units and the height is \(7\) units. Plugging in these values:
\[
A_{\text{triangle}} = \frac{1}{2} \times 24 \times 7 = \frac{168}{2} = 84 \text{ square units}
\]
Since there are two triangular faces, the total area for both triangular faces is:
\[
\text{Total area of triangular faces} = 2 \times 84 = 168 \text{ square units}
\]
2. **Area of the Rectangular Faces:**
The prism has three rectangular faces. We need to calculate the area for each of these rectangular faces.
- Two of the rectangular faces have dimensions of the height of the triangular face and the length of the prism. The height is \(7\) units and the length of the prism is \(100\) units:
\[
A_{\text{rectangular face 1}} = \text{height} \times \text{length} = 7 \times 100 = 700 \text{ square units}
\]
- The third rectangular face has the base of the triangular face and the length of the prism. The base is \(24\) units and the length is \(100\) units:
\[
A_{\text{rectangular face 2}} = \text{base} \times \text{length} = 24 \times 100 = 2400 \text{ square units}
\]
Adding up the areas of all three rectangular faces, we have:
\[
\text{Total area of rectangular faces} = 2 \times 700 + 2400 = 1400 + 2400 = 3800 \text{ square units}
\]
3. **Total Surface Area:**
Now, we sum the areas of the triangular faces and the rectangular faces to find the total surface area of the prism:
\[
\text{Total surface area} = \text{Total area of triangular faces} + \text{Total area of rectangular faces} = 168 + 3800 = 3968 \text{ square units}
\]
Thus, the total surface area of the triangular prism is:
\[
\boxed{3968} \text{ square units}
\]
1. **Area of the Triangular Faces:**
The triangular faces of the prism are isosceles triangles, and the area \(A\) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base of the triangular face is \(24\) units and the height is \(7\) units. Plugging in these values:
\[
A_{\text{triangle}} = \frac{1}{2} \times 24 \times 7 = \frac{168}{2} = 84 \text{ square units}
\]
Since there are two triangular faces, the total area for both triangular faces is:
\[
\text{Total area of triangular faces} = 2 \times 84 = 168 \text{ square units}
\]
2. **Area of the Rectangular Faces:**
The prism has three rectangular faces. We need to calculate the area for each of these rectangular faces.
- Two of the rectangular faces have dimensions of the height of the triangular face and the length of the prism. The height is \(7\) units and the length of the prism is \(100\) units:
\[
A_{\text{rectangular face 1}} = \text{height} \times \text{length} = 7 \times 100 = 700 \text{ square units}
\]
- The third rectangular face has the base of the triangular face and the length of the prism. The base is \(24\) units and the length is \(100\) units:
\[
A_{\text{rectangular face 2}} = \text{base} \times \text{length} = 24 \times 100 = 2400 \text{ square units}
\]
Adding up the areas of all three rectangular faces, we have:
\[
\text{Total area of rectangular faces} = 2 \times 700 + 2400 = 1400 + 2400 = 3800 \text{ square units}
\]
3. **Total Surface Area:**
Now, we sum the areas of the triangular faces and the rectangular faces to find the total surface area of the prism:
\[
\text{Total surface area} = \text{Total area of triangular faces} + \text{Total area of rectangular faces} = 168 + 3800 = 3968 \text{ square units}
\]
Thus, the total surface area of the triangular prism is:
\[
\boxed{3968} \text{ square units}
\]
There are no human answers yet. A form for humans to post answers is coming very soon!