The radius of the circumscribed circle

Text

For a triangle with sides $a,b,c$ and area $A$, the radius of the circumscribed circle is $R=\dfrac{abc}{4A}$. Compute $R$ for the triangle with sides $13$, $14$, $15$.

Solution

First the area with Heron’s formula: $s=\frac{13+14+15}{2}=21$, so $A=\sqrt{21\cdot 8\cdot 7\cdot 6}=\sqrt{7056}=84.$ Then $R=\frac{13\cdot 14\cdot 15}{4\cdot 84}=\frac{2730}{336}=\frac{65}{8}=\boxed{8.125}.$