Question

Build an equation to calculate the surface area of a equilateral triangular prism

Answer 1

To calculate the surface area of an equilateral triangular prism, we need to consider both the lateral surface area and the area of the two triangular bases.

1. **Base Area**: The base of the prism is an equilateral triangle. If the side length of the triangle is \( s \), the area (\( A \)) of one equilateral triangle can be calculated using the formula:
\[
A = \frac{\sqrt{3}}{4} s^2
\]

2. **Lateral Surface Area**: The lateral surface area (the area of the three rectangular sides) can be calculated by multiplying the perimeter of the triangular base by the height (\( h \)) of the prism. The perimeter \( P \) of an equilateral triangle with side length \( s \) is:
\[
P = 3s
\]
Thus, the lateral surface area (\( LSA \)) is:
\[
LSA = P \times h = 3s \times h = 3sh
\]

3. **Total Surface Area**: The total surface area (\( SA \)) of the prism can be found by adding the lateral surface area and the areas of the two bases:
\[
SA = LSA + 2 \times A
\]
Putting it all together:
\[
SA = 3sh + 2 \left( \frac{\sqrt{3}}{4} s^2 \right)
\]

Simplifying the equation gives us:
\[
SA = 3sh + \frac{\sqrt{3}}{2} s^2
\]

Thus, the final equation for the surface area of an equilateral triangular prism is:
\[
SA = 3sh + \frac{\sqrt{3}}{2} s^2
\]