Find the surface area of a right triangular prism if the triangular base has a perpendicular height
of 12 m and a base of 18 m. The height of 12 m and base of 18 m form the two short sides of
the triangular base. The triangular prism has a length of 20 m.

To find the surface area of a right triangular prism, we need to calculate the area of the two triangular bases and the area of the three rectangular sides. Let's break it down step by step:

### 1. **Area of the triangular base**:
The formula for the area of a triangle is:

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

Given:
- Base \( = 18 \, \text{m} \)
- Height \( = 12 \, \text{m} \)

\[
\text{Area of triangle} = \frac{1}{2} \times 18 \, \text{m} \times 12 \, \text{m} = 108 \, \text{m}^2
\]

### 2. **Perimeter of the triangular base**:
To find the area of the rectangular sides, we need the perimeter of the triangular base. First, calculate the hypotenuse of the triangle using the Pythagorean theorem:

\[
\text{Hypotenuse} = \sqrt{(\text{base})^2 + (\text{height})^2} = \sqrt{18^2 + 12^2} = \sqrt{324 + 144} = \sqrt{468} \approx 21.63 \, \text{m}
\]

Now, the perimeter of the triangular base is:

\[
\text{Perimeter} = 18 \, \text{m} + 12 \, \text{m} + 21.63 \, \text{m} = 51.63 \, \text{m}
\]

### 3. **Area of the three rectangular sides**:
The prism has three rectangular faces. The area of each face is given by multiplying the length of the prism by one side of the triangle:
- One rectangle has an area equal to \( \text{base} \times \text{length} = 18 \, \text{m} \times 20 \, \text{m} = 360 \, \text{m}^2 \)
- Another rectangle has an area equal to \( \text{height} \times \text{length} = 12 \, \text{m} \times 20 \, \text{m} = 240 \, \text{m}^2 \)
- The third rectangle has an area equal to \( \text{hypotenuse} \times \text{length} = 21.63 \, \text{m} \times 20 \, \text{m} = 432.6 \, \text{m}^2 \)

### 4. **Total surface area of the prism**:
The total surface area is the sum of the areas of the two triangular bases and the three rectangular sides.

\[
\text{Surface area} = 2 \times \text{Area of triangular base} + \text{Area of three rectangular sides}
\]
\[
\text{Surface area} = 2 \times 108 \, \text{m}^2 + (360 \, \text{m}^2 + 240 \, \text{m}^2 + 432.6 \, \text{m}^2)
\]
\[
\text{Surface area} = 216 \, \text{m}^2 + 1032.6 \, \text{m}^2 = 1248.6 \, \text{m}^2
\]

### Final Answer:
The surface area of the right triangular prism is approximately \( 1248.6 \, \text{m}^2 \).

Would you like more details on any part of the solution or have further questions?

### Related Questions:
1. How would the surface area change if the length of the prism were increased by 10 meters?
2. What would be the volume of the triangular prism with the given dimensions?
3. How would you calculate the surface area if the triangular base were an equilateral triangle?
4. If the hypotenuse were doubled, how would the perimeter and surface area change?
5. What formula would you use to find the surface area of a rectangular prism instead?

### Tip:
When working with prisms, knowing the shape of the base helps you determine how to calculate both the volume and the surface area.

Find the surface area of a right triangular prism if the triangular base has a perpendicular height of 12 m and a base of 18 m. The height of 12 m and base of 18 m form the two short sides of the triangular base. The triangular prism has a length of 20 m.

Calculate the surface area of a right triangular prism with a base of 18 meters, a height of 12 meters, and a prism length of 20 meters. This detailed guide uses the Pythagorean theorem and surface area formulas to provide a complete step-by-step solution, ideal for students in Grades 8-10.

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