Question

Surface area of triangular prism with top and bottom bases of 5, rectangular faces of 6 and length of 8

Answer 1

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

1. **Area of the triangular base**:
The base of the triangular prism is described as a triangle with a base length of 5. The height of this triangle is not given directly, but we can determine it from the other given dimensions. Let's assume the triangular base is an isosceles triangle with the two equal sides being of length 6.

We can use the following formula to find the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
To find the height, we can create a right triangle by dropping a perpendicular from the apex of the triangle to the midpoint of the base. The length of the base is 5, so the half base will be \( \frac{5}{2} = 2.5 \).

Now, we will use the Pythagorean theorem to find the height \( h \):
\[
6^2 = h^2 + 2.5^2
\]
\[
36 = h^2 + 6.25
\]
\[
h^2 = 36 - 6.25 = 29.75
\]
\[
h = \sqrt{29.75} \approx 5.45
\]

Now we can calculate the area of one triangular base:
\[
\text{Area} = \frac{1}{2} \times 5 \times 5.45 \approx 13.625
\]

Since there are 2 triangular bases, the total area of the triangular bases is:
\[
\text{Total area of triangular bases} = 2 \times 13.625 \approx 27.25
\]

2. **Area of the rectangular faces**:
There are three rectangular faces. Each rectangle is calculated as:
- The face connecting the two bases: the length is 8, and the width is the side of the triangle (6).
- The other two faces (each also have a length of 8):
- One of 6 (the other equal side of the triangle),
- And one of 5 (the base of the triangle).

So the areas of the rectangular faces are:
- Rectangular face with side 6:
\[
\text{Area} = 6 \times 8 = 48
\]
- Rectangular face with side 5:
\[
\text{Area} = 5 \times 8 = 40
\]
- The other rectangular face with side 6:
\[
\text{Area} = 6 \times 8 = 48
\]

Now, sum the area of the rectangular faces:
\[
\text{Total area of rectangular faces} = 48 + 40 + 48 = 136
\]

3. **Total surface area**:
Now, add the total area of the triangular bases and the total area of the rectangular faces:
\[
\text{Total surface area} = \text{Total area of triangular bases} + \text{Total area of rectangular faces}
\]
\[
\text{Total surface area} = 27.25 + 136 = 163.25
\]

Thus, the surface area of the triangular prism is approximately **163.25 square units**.