Question

find the surface area of this triangular prism. be sure to include the correct unit in your answer.

Explanation:

Step1: Identify the triangular base and lateral faces

The triangular prism has a right triangular base with legs 5 ft and 12 ft, and the hypotenuse can be found (but we already have the other sides of the prism: length 11 ft, and the other edges 13 ft? Wait, no, let's check the dimensions. The triangular base is a right triangle with legs 5 ft and 12 ft, so area of the triangular base is $\frac{1}{2} \times 5 \times 12$. Then there are two triangular bases, so total area for the two triangles is $2 \times \frac{1}{2} \times 5 \times 12$. Then the lateral faces: the three rectangles. The lengths of the sides of the triangle are 5 ft, 12 ft, and 13 ft (since $5^2 + 12^2 = 25 + 144 = 169 = 13^2$), and the length of the prism (the distance between the two triangular bases) is 11 ft. So the lateral faces are rectangles with dimensions: 5 ft × 11 ft, 12 ft × 11 ft, and 13 ft × 11 ft. Wait, but in the diagram, there's also 11 ft and 13 ft? Wait, maybe I misread. Let's re-express:

The formula for the surface area of a triangular prism is $SA = 2B + Ph$, where $B$ is the area of the triangular base, $P$ is the perimeter of the triangular base, and $h$ is the length of the prism (the distance between the two bases).

First, find $B$: the triangular base is a right triangle with legs 5 ft and 12 ft. So $B = \frac{1}{2} \times 5 \times 12 = 30$ square feet.

Then, the perimeter $P$ of the triangular base: the sides are 5 ft, 12 ft, and 13 ft (since it's a right triangle, hypotenuse is 13). So $P = 5 + 12 + 13 = 30$ feet.

The length of the prism (the distance between the two bases) is 11 ft (from the diagram, the side labeled 11 ft is the length of the prism). Wait, but in the diagram, there's also 11 ft, 13 ft, 12 ft, 5 ft. Wait, maybe the prism has length 11 ft, and the triangular base has sides 5, 12, 13, and the length of the prism is 11. Wait, but let's check the lateral faces. Wait, maybe the three rectangles are 5×11, 12×11, and 13×11? Wait, no, maybe the length of the prism is 11, and the triangular base has legs 5 and 12, hypotenuse 13, and the other edges: wait, the diagram shows 11 ft, 13 ft, 12 ft, 5 ft. Let's confirm:

Wait, the triangular base is a right triangle with legs 5 and 12, hypotenuse 13. Then the prism extends along the 11 ft direction. So the two triangular bases: area each is 30, so 2×30=60.

Then the lateral surface area: the perimeter of the triangle (5+12+13=30) multiplied by the length of the prism (11). Wait, no, the formula $SA = 2B + Ph$ where $h$ is the height of the prism (the distance between the two bases). So if the length of the prism is 11, then $Ph = (5 + 12 + 13) \times 11 = 30 \times 11 = 330$. Then total surface area is $2B + Ph = 60 + 330 = 390$? Wait, but that seems off. Wait, maybe I made a mistake. Wait, let's check the diagram again. The prism has a right triangular base with legs 5 ft and 12 ft, and the other side of the prism (the length) is 11 ft. Then the three lateral faces: one is 5×11, one is 12×11, and one is 13×11? Wait, no, the hypotenuse is 13, so the three rectangles are 5×11, 12×11, and 13×11? Wait, but in the diagram, there's also 11 ft and 13 ft. Wait, maybe the length of the prism is 11, and the triangular base has sides 5, 12, 13, and the prism's length is 11. Then:

Area of two triangular bases: 2 * (0.5 * 5 * 12) = 60.

Lateral surface area: (5 * 11) + (12 * 11) + (13 * 11) = 55 + 132 + 143 = 330.

Total surface area: 60 + 330 = 390. Wait, but let's check again. Wait, maybe the length of the prism is 11, and the triangular base is 5, 12, 13, and the other edges: the height of the prism is 11? Wait, maybe I misread the diagram. Let's re-express:

The triangular prism has a right triangular base with legs 5 ft (base) and 12 ft (height), so the area of one triangle is (5*12)/2 = 30. Then there are two such triangles, so 2*30 = 60.

Then the three rectangular faces:

1. The rectangle with dimensions 5 ft (base of triangle) and 11 ft (length of prism): area = 5*11 = 55.

2. The rectangle with dimensions 12 ft (height of triangle) and 11 ft (length of prism): area = 12*11 = 132.

3. The rectangle with dimensions 13 ft (hypotenuse of triangle) and 11 ft (length of prism): area = 13*11 = 143.

Now, sum the areas of the two triangles and the three rectangles: 60 + 55 + 132 + 143 = 60 + (55 + 132) + 143 = 60 + 187 + 143 = 60 + 330 = 390. So the surface area is 390 square feet.

Wait, but let's confirm the formula. The surface area of a triangular prism is indeed 2 times the area of the triangular base plus the perimeter of the triangular base times the length of the prism (the distance between the two bases). So $SA = 2B + P \times l$, where $l$ is the length of the prism.

Here, $B = \frac{1}{2} \times 5 \times 12 = 30$, $P = 5 + 12 + 13 = 30$, $l = 11$. So $SA = 2*30 + 30*11 = 60 + 330 = 390$. So the surface area is 390 square feet.

Step2: Calculate each part

First, area of the triangular base: $B = \frac{1}{2} \times 5 \times 12 = 30$ $ft^2$.

Two triangular bases: $2B = 2 \times 30 = 60$ $ft^2$.

Perimeter of the triangular base: $P = 5 + 12 + 13 = 30$ $ft$.

Lateral surface area: $P \times l = 30 \times 11 = 330$ $ft^2$ (where $l = 11$ ft is the length of the prism).

Total surface area: $2B + P \times l = 60 + 330 = 390$ $ft^2$.

Answer:

390 $ft^2$