Question

A circle is inscribed in an equilateral triangle of side a. The area of square inscribed inside the circle is:

Answer 1

Hint: Start by drawing the diagram of the whole situation and a diagram of the circle and square separately for better understanding. For finding the radius of the circle use the formula $ r=\left( s-a \right)\tan \dfrac{A}{2} $ and the known thing that s of a equilateral triangle is half of three times of its sides and all angles are equal to $ 60{}^\circ . $ Once you have the radius use the property that the diagonals of the square are equal to the diameter of the circle in which it is inscribed. Also, use the formula that the area of a square is equal to half of the square of its diagonals.

Complete step-by-step answer:

Let us start by drawing the figure of the situation given in the question for better visualisation.

As we can see the circle inscribed in a triangle is the incircle and the inradius is given by:

 $ r=\left( s-a \right)\tan \dfrac{A}{2} $

Also, we know that s of an equilateral triangle is half of three times of its sides and all angles are equal to $ 60{}^\circ . $

 $ r=\left( \dfrac{3}{2}a-a \right)\tan \dfrac{60{}^\circ }{2} $

 $ \Rightarrow r=\dfrac{1}{2}a\times \tan 30{}^\circ $

We know that $ \tan 30{}^\circ =\dfrac{1}{\sqrt{3}} $ .

 $ r=\dfrac{1}{2}a\times \dfrac{1}{\sqrt{3}} $

Now the diameter of the circle is twice its radius, so the length of the diameter is $ 2r=\dfrac{a}{\sqrt{3}} $ . Therefore, the diagonal of the square is also equal to $ 2r=\dfrac{a}{\sqrt{3}} $ .

Also, we know that the area of a square is equal to half of the square of its diagonals.

 $ \therefore \text{ Area of the square=}\dfrac{1}{2}{{\left( diagonal \right)}^{2}} $

 $ \Rightarrow \text{Area of the square=}\dfrac{1}{2}{{\left( \dfrac{a}{\sqrt{3}} \right)}^{2}} $

 $ \text{Area of the square=}\dfrac{1}{2}\times \dfrac{{{a}^{2}}}{3}=\dfrac{{{a}^{2}}}{6}uni{{t}^{2}} $

Therefore, the answer to the above question $ \dfrac{{{a}^{2}}}{6}uni{{t}^{2}} $ .

Note: It is important that you know that the diameter of a circle inscribing a square is equal to the diagonal of the square. Also, remember that the centre of the largest circle that fits inside a triangle is called the incentre and is defined as the meeting point of all three angle bisectors of the triangle.