Calculate the total surface area of the following triangular prism.

Steps to Solve

1. Find the Area of the Triangular Base

To find the area of the triangular base, we can use the formula:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

Here, we can take the base as the side measuring 10 ft and the height corresponds to the line drawn perpendicular from the vertex opposite the base, which is 12 ft.

Calculating the area:

$$ \text{Area}_{\text{triangular base}} = \frac{1}{2} \times 10 \times 12 = 60 , \text{sq ft} $$

2. Find the Area of the Two Triangular Bases

Since there are two triangular bases in the prism, the total area of the triangular bases is:

$$ \text{Total Area}_{\text{triangular bases}} = 2 \times 60 = 120 , \text{sq ft} $$

3. Find the Area of the Rectangular Faces

The prism has three rectangular faces. The areas of these can be found using:

• For the rectangle corresponding to the base 10 ft:

$$ \text{Area} = 10 \times 15 = 150 , \text{sq ft} $$

• For the rectangle corresponding to the side 12 ft:

$$ \text{Area} = 12 \times 15 = 180 , \text{sq ft} $$

• For the rectangle corresponding to the side 13 ft:

$$ \text{Area} = 13 \times 15 = 195 , \text{sq ft} $$

4. Total Rectangular Area Calculation

Adding the areas of the three rectangular faces:

$$ \text{Total Area}_{\text{rectangles}} = 150 + 180 + 195 = 525 , \text{sq ft} $$

5. Calculate the Total Surface Area of the Prism

Finally, the total surface area of the prism is the sum of the areas of the triangular bases and the rectangular faces:

$$ \text{Total Surface Area} = \text{Total Area}{\text{triangular bases}} + \text{Total Area}{\text{rectangles}} $$

Calculating it:

$$ \text{Total Surface Area} = 120 + 525 = 645 , \text{sq ft} $$