Question

The base of a triangular prism is 6 ft. The height is 4 ft. The right length is 5 ft. The width is 8 ft. Find the lateral and surface area of the prism.

Answer 1

To find the lateral and surface area of a triangular prism, we need to know the dimensions of the prism and the area of its base.

1. **Base Area of the Triangular Prism**:
The triangular base has a base \( b = 6 \) ft and a height \( h = 4 \) ft. The area \( A \) of the triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 6 \times 4 = 12 \text{ ft}^2
\]

2. **Lateral Area**:
The lateral area of a prism is the sum of the areas of the three rectangular faces connecting the triangular bases. The lateral area \( LA \) can be calculated using the formula:
\[
LA = \text{Perimeter of base} \times \text{Height of prism}
\]
To find the perimeter of the base triangle, we need the lengths of all three sides. The given right length (hypotenuse) is 5 ft, and one leg is 4 ft (which corresponds to the height). The other leg, which can be calculated using the Pythagorean theorem, is as follows:

Let \( a \) be the leg opposite the 4 ft side:
\[
a^2 + 4^2 = 5^2 \\
a^2 + 16 = 25 \\
a^2 = 9 \\
a = 3 \text{ ft}
\]

Now the lengths of the sides of the triangle are 6 ft, 4 ft, and 5 ft. The perimeter \( P \) of the triangular base is:
\[
P = 6 + 4 + 5 = 15 \text{ ft}
\]

Now we can calculate the lateral area:
\[
LA = P \times \text{Height} = 15 \text{ ft} \times 8 \text{ ft} = 120 \text{ ft}^2
\]

3. **Surface Area**:
The surface area \( SA \) of the prism is the sum of the lateral area and the area of the two triangular bases:
\[
SA = LA + 2 \times \text{Area of triangular base} \\
SA = 120 \text{ ft}^2 + 2 \times 12 \text{ ft}^2 = 120 \text{ ft}^2 + 24 \text{ ft}^2 = 144 \text{ ft}^2
\]

### Summary of Results:
- **Lateral Area**: \( 120 \text{ ft}^2 \)
- **Surface Area**: \( 144 \text{ ft}^2 \)