How do i calculate the area of shaded region?

Question

I wouldn like to find the area of shaded region which it's circulated by a triangle as show in the below picture ?

Note: I tried to draw other circle arround triangle ,but it's seems hard to me to deduce the shaded area

Thank you for any help

Answer 1

Let the area of the triangle be $\Delta$, and the inradius be $r$ Then dividing the triangle into three by joining each vertex to the incentre we get $r(a+b+c)=2\Delta$. The area of the circle is $\pi r^2$. So all we need is the formula for $\Delta$ which is $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=(a+b+c)/2$.

With $a=15,b=14,c=13$, we have $s=21$, so $\Delta=84$ and hence $r=168/42=4$. So the shaded area is $84-16\pi\approx33.73$.