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this is an image of an equilateral triangular prism. what is the surface area?

To calculate the surface area of an equilateral triangular prism, we need to determine the area of its three rectangular faces and the two triangular bases. The given dimensions are:

- Base length of the triangle, \( b = 12 \) cm
- Height of the triangle, \( h = 10.4 \) cm
- Length of the prism, \( l = 91.5 \) cm

Since the prism is equilateral, all sides of the triangular base are equal.

### Steps to Calculate the Surface Area

1. **Area of one triangular base:**
   The area \( A \) of an equilateral triangle can be calculated using the formula:
   \[
   A = \frac{1}{2} \times b \times h
   \]
   Plugging in the values:
   \[
   A = \frac{1}{2} \times 12 \text{ cm} \times 10.4 \text{ cm} = 62.4 \text{ cm}^2
   \]
   Since there are two triangular bases:
   \[
   \text{Total area of the triangular bases} = 2 \times 62.4 \text{ cm}^2 = 124.8 \text{ cm}^2
   \]

2. **Area of the three rectangular faces:**
   Each rectangular face has a length of 91.5 cm and a width equal to the side length of the triangular base (12 cm). The area \( A_{rect} \) of one rectangle is:
   \[
   A_{rect} = l \times b
   \]
   Plugging in the values:
   \[
   A_{rect} = 91.5 \text{ cm} \times 12 \text{ cm} = 1098 \text{ cm}^2
   \]
   Since there are three rectangular faces:
   \[
   \text{Total area of the rectangular faces} = 3 \times 1098 \text{ cm}^2 = 3294 \text{ cm}^2
   \]

3. **Total surface area of the prism:**
   The total surface area \( SA \) is the sum of the areas of the triangular bases and the rectangular faces:
   \[
   SA = 124.8 \text{ cm}^2 + 3294 \text{ cm}^2 = 3418.8 \text{ cm}^2
   \]

So, the surface area of the equilateral triangular prism is \( 3418.8 \text{ cm}^2 \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you calculate the volume of an equilateral triangular prism?
2. What is the relationship between the base area and the height of an equilateral triangle?
3. How would the surface area change if the prism were not equilateral?
4. Can you derive the height of an equilateral triangle given its side length?
5. How do you find the lateral surface area of a triangular prism?
6. What is the formula for the surface area of a right triangular prism?
7. How does the length of the prism affect the total surface area?
8. How do you determine the length of a prism if you know its surface area and base dimensions?

**Tip:** When dealing with prisms and other 3D shapes, remember that surface area includes all the outer surfaces, while volume measures the space contained within the shape.

Learn how to calculate the surface area of an equilateral triangular prism with a detailed step-by-step solution. This problem covers the area formulas for equilateral triangles and rectangles. Suitable for advanced high school students.

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