Question

let me see who is here a genius to solve this problem ​

Answer 1

Answer:

Area of a circle inscribed in a regular hexagon. Given a regular Hexagon with side length a, the task is to find the area of the circle inscribed in it, given that, the circle is tangent to each of the six sides.

Step-by-step explanation:

Given a regular Hexagon with side length a, the task is to find the area of the circle inscribed in it, given that, the circle is tangent to each of the six sides.

We can divide the regular hexagon into 6 identical equilateral triangles.

We take one triangle OAB, with O as the centre of the hexagon or circle, & AB as one side of the hexagon.

Let M be mid-point of AB, OM would be the perpendicular bisector of AB, angle AOM = 30 deg

Then in right angled triangle OAM,

tanx = tan30 = 1/√3

So, a/2r = 1/√3

Therefore, r = a√3/2

Area of circle, A =Πr²=Π3a^2/4