Sum of Sides of Triangle in terms of Circumradius and Half Angle Cosines

Theorem
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.

Then:
 * $a + b + c = 8 R \cos \dfrac A 2 \cos \dfrac B 2 \cos \dfrac C 2$

where $R$ denotes the circumradius of $\triangle ABC$.

Proof
From Law of Sines:


 * $\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$

Hence: