Question

A ramp is in the shape of a triangular prism the ramp and its net shown below what is the surface area of the ramp 5ft, 6ft, 13ft,12ft

Answer 1

To find the surface area of a triangular prism, we need to determine the areas of both the triangular bases and the rectangular sides.

Given dimensions:
- The base of the triangular base is 5 ft.
- The height of the triangular base is 12 ft.
- The lengths of the prism (the sides of the triangular prism) are 6 ft and 13 ft.

### Step 1: Find the area of the triangular base.

The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Plugging in the values for the triangular base:
\[
A = \frac{1}{2} \times 5 \, \text{ft} \times 12 \, \text{ft} = \frac{1}{2} \times 60 = 30 \, \text{ft}^2
\]

### Step 2: Find the area of the two triangular bases.

Since there are two triangular bases, the total area of the bases is:
\[
\text{Total area of triangular bases} = 2 \times A = 2 \times 30 \, \text{ft}^2 = 60 \, \text{ft}^2
\]

### Step 3: Find the area of the rectangular sides.

The prism has three rectangular sides. The area of each rectangular side can be determined as follows:

1. Rectangular side with length 6 ft and height of 12 ft:
\[
A_1 = \text{height} \times \text{length} = 12 \, \text{ft} \times 6 \, \text{ft} = 72 \, \text{ft}^2
\]

2. Rectangular side with length 5 ft and height of 12 ft:
\[
A_2 = 12 \, \text{ft} \times 5 \, \text{ft} = 60 \, \text{ft}^2
\]

3. Rectangular side with length 13 ft and height of 12 ft:
\[
A_3 = 12 \, \text{ft} \times 13 \, \text{ft} = 156 \, \text{ft}^2
\]

### Step 4: Calculate the total surface area of the prism.

Now, add up all of the areas:
\[
\text{Total surface area} = \text{Area of triangular bases} + \text{Area of rectangular sides}
\]
\[
= 60 \, \text{ft}^2 + (A_1 + A_2 + A_3) = 60 \, \text{ft}^2 + (72 \, \text{ft}^2 + 60 \, \text{ft}^2 + 156 \, \text{ft}^2)
\]
Calculating the total area of the rectangular sides:
\[
A_1 + A_2 + A_3 = 72 + 60 + 156 = 288 \, \text{ft}^2
\]
Now combine with the triangular bases:
\[
\text{Total surface area} = 60 + 288 = 348 \, \text{ft}^2
\]

Thus, the surface area of the ramp is **348 square feet**.