Question

if the perimeter of an equilateral triangle is 72 m,then its area is

Answer 1

Task Suppose the perimeter of an equilateral triangle is 72 m. Find its area.

Answer. Equilateral triangle is the triangle all of whose sides are equal. Let a be the side of that triangle, then

$P=a+a+a=3a=72$

whence

$a=\frac{72}{3}=24m.$

Recall that the area of a triangle with sides $a$, $b$ and the angle $\alpha$ between them is equal to

$A=\frac{1}{2}ab\sin\alpha.$

The angle between any two sides of such an equilateral triangle is $\frac{\pi}{3}$, whence its area is equal to

$A=\frac{1}{2}a\cdot a\cdot\sin\frac{\pi}{3}=\frac{1}{2}\cdot 24\cdot 24\cdot\frac{\sqrt{3}}{2}=144\sqrt{3}m^{2}.$