Question

Surface area of triangular prism
8cm
10cm
6cm
9cm

Answer 1

To find the surface area of a triangular prism, you'll need to calculate both the lateral surface area and the area of the triangular bases.

1. **Triangular Base Area:**
The formula for the area of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we need to determine which sides of the triangle are the base and the height. Let's assume that \(b = 6 \, \text{cm}\) (base), and to find the height \(h\), we consider the 8 cm and 10 cm sides as the other sides of the triangle.

We can use Heron's formula to find the area when we just have the lengths of the sides. The sides of the triangle are 8 cm, 6 cm, and 10 cm.

First, calculate the semi-perimeter \(s\):
\[
s = \frac{a + b + c}{2} = \frac{8 + 6 + 10}{2} = 12 \, \text{cm}
\]

Now, use Heron's formula for the area \(A\):
\[
A = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{12(12 - 8)(12 - 6)(12 - 10)}
\]
\[
= \sqrt{12 \times 4 \times 6 \times 2} = \sqrt{576} = 24 \, \text{cm}^2
\]

2. **Lateral Surface Area:**
The lateral surface area of the prism is given by:
\[
\text{Lateral Surface Area} = \text{Perimeter of the base} \times \text{Height}
\]
The perimeter of the triangular base is:
\[
P = 8 + 6 + 10 = 24 \, \text{cm}
\]
Assume the height of the prism (length) is \(h = 9 \, \text{cm}\).

Thus, the lateral surface area is:
\[
\text{Lateral Surface Area} = 24 \times 9 = 216 \, \text{cm}^2
\]

3. **Total Surface Area:**
Finally, the total surface area \(SA\) of the prism is the sum of the lateral surface area and the area of the two triangular bases:
\[
SA = \text{Lateral Surface Area} + 2 \times \text{Area of base}
\]
Therefore:
\[
SA = 216 + 2 \times 24 = 216 + 48 = 264 \, \text{cm}^2
\]

The total surface area of the triangular prism is \( \text{264 cm}^2 \).