Question

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

9-inch perimeter

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 = √(AC^2-AH^2)=\sqrt{l^2-(l^2)/(4)}=(√(3)l)/(2)`

So, the area is

`(AH\cdot HC)/(2) = (l\cdot(√(3)l)/(2))/(2) = (l^2√(3))/(4) = (9√(3))/(4)`