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What is the total surface area of the triangular prism shown?

The figure you provided is of a triangular prism. To calculate the total surface area of this triangular prism, we need to find the areas of all the faces and sum them up. The surface area of a prism is calculated as:

\[
\text{Total Surface Area} = 2 \times \text{Base Area} + \text{Lateral Surface Area}
\]

Where:
- The "Base Area" is the area of one of the triangular faces.
- The "Lateral Surface Area" is the sum of the areas of the rectangular faces that connect the triangular bases.

### Step 1: Finding the area of the triangular base
The given dimensions form a right triangle with legs of 9 in and 12 in, and the hypotenuse is not necessary for the area of the triangle. The area of a right triangle is given by:

\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here, the base = 9 inches and the height = 12 inches, so:

\[
\text{Area of triangle} = \frac{1}{2} \times 9 \times 12 = 54 \, \text{square inches}
\]

### Step 2: Finding the lateral surface area
The lateral faces are rectangular, and their areas are calculated by multiplying the length of the prism (15 inches) by the corresponding side of the triangular base. There are three lateral faces:

1. \( 15 \, \text{inches} \times 9 \, \text{inches} \)
2. \( 15 \, \text{inches} \times 12 \, \text{inches} \)
3. \( 15 \, \text{inches} \times 10 \, \text{inches} \) (the hypotenuse)

The areas are:
1. \( 15 \times 9 = 135 \, \text{square inches} \)
2. \( 15 \times 12 = 180 \, \text{square inches} \)
3. \( 15 \times 10 = 150 \, \text{square inches} \)

### Step 3: Summing the areas
Now, we sum the areas of all the faces:

- Base Area = \( 2 \times 54 = 108 \, \text{square inches} \) (since there are two triangular bases)
- Lateral Surface Area = \( 135 + 180 + 150 = 465 \, \text{square inches} \)

### Step 4: Total Surface Area
\[
\text{Total Surface Area} = 108 + 465 = 573 \, \text{square inches}
\]

Thus, the total surface area of the triangular prism is **573 square inches**.

Would you like further clarification or additional details on any part of this solution? Here are some related questions for deeper understanding:

1. How is the area of a right triangle calculated in general?
2. What would change if the triangular base wasn't a right triangle?
3. Can you apply this formula to other types of prisms, like a rectangular prism?
4. How would the volume of this prism be calculated?
5. How do the lateral surface areas depend on the shape of the base?

**Tip:** Always ensure to verify whether a triangle is a right triangle when using the basic formula for its area—it simplifies many calculations!

This problem involves finding the surface area of a triangular prism with a right triangular base. The given dimensions of the triangle are 9 inches and 12 inches for the legs, with a 15-inch length for the prism. The surface area is calculated by finding the area of the triangular base and adding the areas of the lateral faces. The total surface area is 573 square inches.

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