Question

In the given figure, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region.
[Use $\sqrt{3}=1.73,\mathrm{\pi }=3.14$] [CBSE 2014]

Answer 1

We can find the radius of the incircle by using the formula
$r=\frac{2\times \mathrm{Area}\mathrm{of}\mathrm{triangle}}{\mathrm{Perimeter}\mathrm{of}\mathrm{triangle}}=\frac{2\times {\displaystyle \frac{\sqrt{3}}{4}}\times {\left(12\right)}^2}{3\times 12}=2\sqrt{3}\mathrm{cm}$
Now, area of shaded region = Area of triangle − Area of circle
$$\begin{gathered} =\frac{\sqrt{3}}{4}\times {\left(12\right)}^2-3.14\times {\left(2\sqrt{3}\right)}^2 \\ =62.28-37.68 \\ =24.6{\mathrm{cm}}^2 \end{gathered}$$
Hence, the area of shaded region is 24.6 cm²