Question

ABCDEF is a regular hexagon. With vertices A, B, C, D, E and F as the centres, circles of same radius ‘r’ are drawn. Find the area of the shaded portion shown in the given figure.

Answer 1

FIGURE IS IN THE ATTACHMENT.
Firstly we need to find the angle of regular hexagon.
Each angle of a regular hexagon= sum of all angles / number of sides.
Each angle of a regular hexagon= (6-2)×180° / 6 = 4 ×30= 120°
[sum of all angles =( n -2)×180°]

∠A = ∠B = ∠C = ∠D = ∠E = ∠F= 120°

GIVEN:
Radius of Circle = r

Area of sector with Central angle A = (∠A /360°) × πr²
= (120° /360°)/πr² = ⅓(πr² )

Area of sector with Central angle A = ⅓(πr² )

Since all angles of a regular hexagon are equal.
Area of shaded region = 6 × Area of sector with Central angle A
Area of shaded region = 6 × ⅓(πr²) = 2πr²

Hence, the Area of shaded portion = 2πr²

HOPE THIS WILL HELP YOU.....