# Circle Circumscribed around triangle, find radius

The question: Find the radius of the circle circumscribed around a triangle with side lengths 7, 9, 12.

Using the cosine law, I found the three angles to be 35.43094, 96. 37937, and 48.18969. However, I am stuck on how I can use the angles and side lengths to find the radius of the circles.

The full statement of the Sine Rule is $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac {c}{\sin C}=\color{red}{2R}$$
Where $R$ is the radius of the circumcircle
$s=\dfrac{7+9+12}2=14$ is semiperimeter. By Heron formula $\text{Area}(ABC)=\sqrt{14 \cdot 7\cdot 5 \cdot 2}= 14 \sqrt5$. If circumradius of $\triangle ABC$ is $R$, then $$\text{Area}(ABC)=\dfrac{abc}{4R}$$ Therefore we find that $R=\dfrac{27\sqrt5}{10}$.