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

enter image description here

Thank you for any help


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 $84-16\pi\approx33.73$.


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