"A net of a triangular prism is drawn unfolded and open. It shows a dashed line equilateral triangle in the center, with 3 triangles adjoined to the baselines of the central triangle.

Question

A baseline of the central triangle is labeled as 17 feet. The perpendicular height of the central base triangle is 14.7 feet, drawn with a dotted line and a right angle symbol with the base. The perpendicular height of an adjoined triangle is shown as 13.8 feet, and is drawn with a dotted line and a right angle symbol with the base."/>
whats the surface area

Answer 1

To find the surface area of the triangular prism, we need to calculate the area of the two triangular bases and the three rectangular faces.

Step 1: Area of the triangular bases

The triangular base is an equilateral triangle with a baseline (side length) of 17 feet. The area \( A \) of an equilateral triangle can be calculated using the formula:

\[ A = \frac{\sqrt{3}}{4} s^2 \]

where \( s \) is the side length of the triangle.

\[ A = \frac{\sqrt{3}}{4} (17)^2 = \frac{\sqrt{3}}{4} \times 289 \approx 125.0 \text{ square feet} \]

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

\[ \text{Total area of bases} = 2 \times A \approx 2 \times 125.0 = 250.0 \text{ square feet} \]

Step 2: Area of the rectangular faces

Each rectangular face has one side equal to the corresponding base of the triangle (17 feet) and one side equal to the height of the triangular prism.

1. Area of the first rectangle (adjoint to the first side of the base)

• Height = 14.7 feet (perpendicular height of the central triangle)
• Area = base × height = 17 feet × 14.7 feet = 250.0 square feet

2. Area of the second rectangle (adjoint to the second side of the base)

• Height = 13.8 feet (perpendicular height of the adjoining triangle)
• Area = base × height = 17 feet × 13.8 feet = 234.6 square feet

3. Area of the third rectangle (adjoint to the final side of the base)

Since the problem description does not specify another height, we assume that the same height as the first rectangle applies (using the central triangle's height).

• Area = 17 feet × 14.7 feet = 250.0 square feet

Total area of the rectangular faces:

\[ \text{Total area of rectangles} = 250.0 + 234.6 + 250.0 \approx 734.6 \text{ square feet} \]

Step 3: Total surface area

Now, we can add the total area of the triangular bases to the total area of the rectangular faces to find the total surface area of the prism.

\[ \text{Total surface area} = \text{Total area of bases} + \text{Total area of rectangles} \]

\[ \approx 250.0 + 734.6 = 984.6 \text{ square feet} \]

Conclusion

The surface area of the triangular prism is approximately 984.6 square feet.