Question
An illustration shows a net version of a triangular prism. The net model shows 3 rectangles top to bottom. Identical triangles are attached to the left and right of the middle rectangle. Parts are labeled with measurements. The top and bottom rectangles have a length measuring 10 centimeters and a width measuring 5 centimeters. The perpendicular height from the top vertex of the triangle to the base measures 4 centimeters. The base of the triangle measures 5 centimeters. The base shares an edge with the middle rectangle.
What is the surface area of the net of the triangular prism?
What is the surface area of the net of the triangular prism?
Answers
GPT-4o mini
To calculate the surface area of the net of the triangular prism, we need to find the area of all the individual faces represented in the net.
### The Components of the Net:
1. **Rectangles (Top and Bottom):**
- Each rectangle has a length of 10 cm and a width of 5 cm.
- Area of one rectangle = length × width = 10 cm × 5 cm = 50 cm².
- Since there are two identical rectangles (top and bottom), the total area for the rectangles is:
\[
2 \times 50 \, \text{cm}^2 = 100 \, \text{cm}^2.
\]
2. **Middle Rectangle:**
- The middle rectangle also has a length of 10 cm and a width of 5 cm.
- Area of the middle rectangle = length × width = 10 cm × 5 cm = 50 cm².
3. **Triangular Faces (Left and Right):**
- Each triangle has a base of 5 cm and a height of 4 cm.
- Area of one triangle = \(\frac{1}{2} \times \text{base} \times \text{height}\) = \(\frac{1}{2} \times 5 \, \text{cm} \times 4 \, \text{cm} = 10 \, \text{cm}^2\).
- Since there are two identical triangles (left and right), the total area for the triangles is:
\[
2 \times 10 \, \text{cm}^2 = 20 \, \text{cm}^2.
\]
### Total Surface Area:
Now we sum up the areas of all the components:
\[
\text{Total Surface Area} = \text{Area of Rectangles} + \text{Area of Middle Rectangle} + \text{Area of Triangles}
\]
\[
= 100 \, \text{cm}^2 + 50 \, \text{cm}^2 + 20 \, \text{cm}^2 = 170 \, \text{cm}^2.
\]
Thus, the total surface area of the net of the triangular prism is **170 cm²**.
### The Components of the Net:
1. **Rectangles (Top and Bottom):**
- Each rectangle has a length of 10 cm and a width of 5 cm.
- Area of one rectangle = length × width = 10 cm × 5 cm = 50 cm².
- Since there are two identical rectangles (top and bottom), the total area for the rectangles is:
\[
2 \times 50 \, \text{cm}^2 = 100 \, \text{cm}^2.
\]
2. **Middle Rectangle:**
- The middle rectangle also has a length of 10 cm and a width of 5 cm.
- Area of the middle rectangle = length × width = 10 cm × 5 cm = 50 cm².
3. **Triangular Faces (Left and Right):**
- Each triangle has a base of 5 cm and a height of 4 cm.
- Area of one triangle = \(\frac{1}{2} \times \text{base} \times \text{height}\) = \(\frac{1}{2} \times 5 \, \text{cm} \times 4 \, \text{cm} = 10 \, \text{cm}^2\).
- Since there are two identical triangles (left and right), the total area for the triangles is:
\[
2 \times 10 \, \text{cm}^2 = 20 \, \text{cm}^2.
\]
### Total Surface Area:
Now we sum up the areas of all the components:
\[
\text{Total Surface Area} = \text{Area of Rectangles} + \text{Area of Middle Rectangle} + \text{Area of Triangles}
\]
\[
= 100 \, \text{cm}^2 + 50 \, \text{cm}^2 + 20 \, \text{cm}^2 = 170 \, \text{cm}^2.
\]
Thus, the total surface area of the net of the triangular prism is **170 cm²**.