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 three rectangular lateral faces. Let's go step-by-step:

### Step 1: Find the Area of the Triangular Base

The triangular base is a right triangle with a perpendicular height of 12 m and a base of 18 m. The area \( A \) of a right triangle is given by:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Substituting the given values:

\[
A = \frac{1}{2} \times 18 \times 12 = \frac{1}{2} \times 216 = 108 \, \text{m}^2
\]

### Step 2: Find the Perimeter of the Triangular Base

To find the lateral surface area, we need the perimeter of the triangular base. The perimeter \( P \) of the triangle is the sum of all its sides. Since it is a right triangle with sides of 18 m and 12 m, the hypotenuse \( c \) can be found using the Pythagorean theorem:

\[
c = \sqrt{18^2 + 12^2} = \sqrt{324 + 144} = \sqrt{468}
\]

Let's compute the exact value:

\[
c = \sqrt{468} \approx 21.63 \, \text{m}
\]

Now, the perimeter \( P \) is:

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

### Step 3: Find the Lateral Surface Area

The lateral surface area of the prism consists of three rectangles. The height (or length) of the prism is given as 20 m. Thus, the lateral surface area \( L \) is:

\[
L = P \times \text{length} = 51.63 \times 20 = 1032.6 \, \text{m}^2
\]

### Step 4: Find the Total Surface Area

The total surface area \( S \) of the prism is the sum of the areas of the two triangular bases and the lateral surface area:

\[
S = 2 \times A + L
\]

Substituting the values:

\[
S = 2 \times 108 + 1032.6 = 216 + 1032.6 = 1248.6 \, \text{m}^2
\]

### Conclusion

The surface area of the right triangular prism is:

\[
\boxed{1248.6 \, \text{m}^2}
\]

Would you like more details on any step or have any other questions? Here are some related questions:

1. How do you find the volume of a triangular prism?
2. What is the Pythagorean theorem, and how is it used to find the hypotenuse?
3. How do you calculate the surface area of other types of prisms?
4. What are some real-life applications of calculating the surface area of a prism?
5. How do you determine the surface area of a triangular prism with different dimensions?

**Tip:** When calculating areas and perimeters of geometric shapes, always ensure your units are consistent and correctly labeled!

Learn how to calculate the surface area of a right triangular prism with a base of 18 m, a perpendicular height of 12 m, and a length of 20 m. This detailed solution includes key geometry concepts, the Pythagorean theorem, and formulas for surface area. Suitable for students in Grades 8-10.

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