Question

solve for surface area of triangular prism
length of 6
width of of 10
height of 8

Answer 1

To find the surface area of a triangular prism, we need to consider the areas of the triangular bases and the three rectangular lateral faces.

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

Given:
- Width (base of triangle) = 10 units
- Height of the triangle = 8 units

The area \( A \) of the triangular base can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values:
\[
A = \frac{1}{2} \times 10 \times 8 = \frac{80}{2} = 40 \text{ square units}
\]

### Step 2: Calculate the lateral surface area.

The prism has three rectangular lateral faces. The dimensions for these rectangles are:
1. One rectangle using the length of the prism and the base of the triangle.
2. One rectangle using the length of the prism and the height of the triangle (from a vertex to the base).
3. One rectangle using the length of the prism and the slant height of the triangle.

To determine the slant height \( s \) of the triangle, we can use the Pythagorean theorem. Assuming the triangular prism has a right triangle at the base:

- Half of the base = \( \frac{10}{2} = 5 \)
- Height = 8

The slant height \( s \) can be found as follows:
\[
s = \sqrt{(\text{height})^2 + \left(\frac{\text{base}}{2}\right)^2} = \sqrt{8^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89} \approx 9.43
\]

Now we calculate the areas of the rectangular faces:
1. First rectangle (base × length):
\[
A_1 = \text{base} \times \text{length} = 10 \times 6 = 60 \text{ square units}
\]
2. Second rectangle (height × length):
\[
A_2 = \text{height} \times \text{length} = 8 \times 6 = 48 \text{ square units}
\]
3. Third rectangle (slant height × length):
\[
A_3 = s \times \text{length} = \sqrt{89} \times 6 \approx 9.43 \times 6 \approx 56.58 \text{ square units}
\]

### Step 3: Calculate the total surface area.

The total surface area \( SA \) is given by:
\[
SA = 2 \times A + A_1 + A_2 + A_3
\]
Substituting the values:
\[
SA = 2 \times 40 + 60 + 48 + 56.58
\]
\[
SA = 80 + 60 + 48 + 56.58 \approx 244.58 \text{ square units}
\]

### Final Answer:
The surface area of the triangular prism is approximately **244.58 square units**.