Question

in fig ABCDEF is any regular hexagon with different vertices A,B,C,D,E,F as the centre's of circles with same radius 'r' are drawn find the shaded region

Answer 1

Let number of sides of a regular polygen is n
∴n× each angle = (n − 2)×180°
For regular hexagon n = 6
∴6× each angle = 4×180°
each angle =120°
Area of 6 shaded regions
6×120/360×pie R2= 2pie R2

Answer 2

Given:

Each angle of the regular hexagon has the same radius with different vertices.

To Find:

The shaded region.

Solution:

Each angle of the regular hexagon is $\frac{Sum of all angles}{Number of sides}$

The intercepts will be 120° for every circle. The Sum of all the angles will be (6-2) × 180° and the number of sides is 6

So, substituting the following

⇒ $\frac{Sum of all angles}{Number of sides}$

⇒ $\frac{(6-2)180}{6}$

Now, we solve

We get,⇒ $\frac{4*180}{6}$

⇒ 120°

∴ ∠A = ∠B = ∠C = ∠D = ∠E = ∠F = 120°

Now, we find the area of the shaded area = 6 × Area of a sector of ∠A

The formula to calculate sector is θ/360πr²

Now, we substitute the values

⇒ 6 × 120/360 × πr²

Further after dividing and multiplying we get 2πr²

Therefore, the area of the shaded area is 2πr².