Question

A circle is inscribed in an equilateral triangle with side 8cm. What will be the area of the portion between the triangle and the circle?

Answer 1

$\begin{array}{c}\frac{r}{4}=\tan {30}^{\circ }\\ \frac{r}{4}=\frac{1}{\sqrt{3}}\\ r=\frac{4}{\sqrt{3}}cm\end{array}$

Area of equilateral triangle is

$\begin{array}{c}\frac{\sqrt{3}}{4}{\left(side\right)}^2\\ =\frac{\sqrt{3}}{4}\times 8\times 8=16\sqrt{3}c{m}^2\end{array}$

Area of circle $=\pi r^2=\pi \times \frac{16}{3}=\frac{\pi }{3}\times 16c{m}^2$

Then, area between triangle and circle is $=\left(16\sqrt{3}-\frac{16\pi }{3}\right)=16\left(\frac{3\sqrt{3}-\pi }{3}\right)c{m}^2$