Question

A circle of radius a is divided into 6 equal sectors. An equilateral triangle is drawn on the chord of each sector to lie outside the circle. Area of the resulting figure is:

Answer 1

Given that, a circle of radius $a$ is divided into $6$ equal sectors. An equilateral triangle is drawn on the chord of each sector to lie outside the circle.

To find out: Area of the resulting figure.

Based on the given information, we can draw the figure shown above.

$△AOB,$ is an equilateral $△$

$\Rightarrow AB=aunits.$

Similarly, $BC=CD=DE=EF=AF=a$

An equilateral $△$ is formed in each sector outside the circle.

Also, the triangles formed inside the sector are equilateral.

$\therefore$ Area of the resultant figure $=12\times Areaofequilateral△$

We know that, the area of an equilateral triangle of side $a$ is $\frac{\sqrt{3}}{4}{a}^2$

$\therefore$ The required area $=12\times \frac{\sqrt{3}}{4}{a}^2$.

$\Rightarrow 3\sqrt{3}{a}^2$

Hence, the area of the resulting figure is $3\sqrt{3}{a}^{2}squareunits$.