The vertices of an equilateral triangle, with perimeter P and area A, lie on a circle with radius r. Find an expression for $P/A$.

Question

The solution to the problem states that the side length of the triangle is $2r \cos(30°)$, multiplies by $3$ to get the perimeter, and then goes on to find the area by applying the formula: $3/2r^2\sin(120°)$.

I don't understand this solution. I don't understand how one can arrive at the value for the side length nor the area (how is the angle $120°?$). I'd appreciate if someone could walk me through this solution.

Answer 1

Let $a$ be the side of the triangle. According to sine law: $\frac{a}{\sin 60°}=2r$ or $a=2r \sin 60°$. Note that $\sin 60°= \cos 30°$. Area of the triangle $A=\frac{a^2\sin 60°}{2}$. This formula is useful when you know two sides and an angle between them.