Question

Find the area of an equilateral triangle (regular 3-gon) with the given measurement.

Answer 1

If the perimeter is 9 inches, each side is 3 inches.

The height of an equilateral triangle splits the triangle in two right sub-triangles: for example, consider a triangle ABC and the height AH starting from A, perpendicular to BC.

The triangle AHC is a right triangle, whose legs are AH and HC, and whose hypotenuse is AC.

AH is half the side, and AC is a side. So, we can use the Pythagorean theorem to find the height:

$AH = \sqrt{AC^2-AH^2}=\sqrt{l^2-\dfrac{l^2}{4}}=\dfrac{\sqrt{3}l}{2}$

So, the area is

$\dfrac{AH\cdot HC}{2} = \dfrac{l\cdot\frac{\sqrt{3}l}{2}}{2} = \dfrac{l^2\sqrt{3}}{4} = \dfrac{9\sqrt{3}}{4}$