Area of a circumscribed circle given sidelengths of inscribed triangle

Question

The sides of a triangle have legths of 7, 15, and 20. What is the exact area of the circumscribed circle?

Answer 1

Let $$ a=7, \quad b=15, \quad c=20. $$ By the law of cosines, $$ a^2 = b^2 + c^2 - 2 b c \cos A, $$ so $$ \cos A = \frac{b^2 + c^2-a^2}{2 b c} = {24\over25}, \qquad \sin A = \sqrt{1-\cos^2A} = {7\over25} $$ From the law of sines, we can find the circumradius $R$: $$ {a\over\sin A} = {b\over\sin B} = {c\over\sin C} = 2R \quad\Rightarrow\quad R = {a\over2\sin A}=12.5 $$ So the area of the circumscribed circle is $$ \pi R^2 = \pi \left( {a\over2\sin A} \right)^2 = {625\over4}\pi. $$