Question

How do I calculate the surface area of a 3D triangular prism?

Answer 1

To calculate the surface area of a 3D triangular prism, you'll need to consider both the triangular bases and the three rectangular sides. Here's a step-by-step guide:

### Step 1: Identify the dimensions
1. **Base Triangle**: Determine the dimensions of the triangular base. Let:
- \( a \), \( b \), and \( c \) be the lengths of the sides of the triangle.
- \( h \) be the height of the triangle.

2. **Height of the Prism**: Let \( H \) be the height (length) of the prism.

### Step 2: Calculate the area of the triangular base
The area \( A_\triangle \) of the triangular base can be calculated using the formula:
\[
A_\triangle = \frac{1}{2} \times \text{base} \times \text{height}
\]
You can also use Heron’s formula if you have all three sides:
\[
s = \frac{a + b + c}{2} \quad (\text{semi-perimeter})
\]
\[
A_\triangle = \sqrt{s(s - a)(s - b)(s - c)}
\]

### Step 3: Calculate the areas of the rectangular sides
The prism has three rectangular sides. The areas of these rectangles are calculated as follows:
- For the side corresponding to side \( a \):
\[
A_a = a \times H
\]
- For the side corresponding to side \( b \):
\[
A_b = b \times H
\]
- For the side corresponding to side \( c \):
\[
A_c = c \times H
\]

### Step 4: Calculate the total surface area
The total surface area \( SA \) of the triangular prism comprises:
- The areas of the two triangular bases (there are two bases):
\[
2 \times A_\triangle
\]
- The areas of the three rectangular sides:
\[
A_a + A_b + A_c
\]

Thus, the total surface area can be expressed as:
\[
SA = 2 \times A_\triangle + A_a + A_b + A_c
\]

### Putting it all together

The complete formula for the surface area of a triangular prism becomes:
\[
SA = 2 \times A_\triangle + (a + b + c) \times H
\]

Make sure to substitute the appropriate values to get your final surface area!