Question

Solve for the surface area of the triangle prism 6cm 10cm 8cm 10cm 10cm

Answer 1

To find the surface area of a triangular prism, we need to know the dimensions of both the triangular base and the height of the prism.

From your message, it seems like we have:

- The lengths of the sides of the triangular base: \( a = 6 \, \text{cm} \), \( b = 8 \, \text{cm} \), and \( c = 10 \, \text{cm} \).
- The height of the prism: \( h = 10 \, \text{cm} \).

### Step 1: Calculate the area of the triangular base

To find the area of the triangle, we can use Heron's formula:

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

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

### Step 2: Calculate the surface area of the prism

The surface area \( SA \) of the triangular prism can be calculated using the formula:
\[
SA = 2A + P \cdot h
\]
Where \( P \) is the perimeter of the triangular base.

1. Calculate the perimeter \( P \):
\[
P = a + b + c = 6 + 8 + 10 = 24 \, \text{cm}
\]

2. Now substitute the values into the surface area formula:
\[
SA = 2 \times A + P \cdot h = 2 \times 24 + 24 \times 10
\]
\[
= 48 + 240 = 288 \, \text{cm}^2
\]

### Final Answer

The surface area of the triangular prism is \( 288 \, \text{cm}^2 \).