Question

Four equal circles, each of radius a units, touch each other. Show that the area between them is `(6/7"a"^2)` sq units.

Answer 1

When four circles touch each other, their centres form the vertices of a square. The sides of the square are 2a units.

Area of the square = (2a)² = 4a² sq. units

Area occupied by the four sectors

`= 4xx90/360xxpixx"a"^2`

= πa²  sq. units

Area between the circles = Area of the square - Area of the four sectors 

`=4xx90xx360xx"a"^2`

= πa² sq. units

Area between the circles = Area of the square -- Area of the four sectors

`= (4 - 22/7)"a"^2`

`=6/7"a"^2   "sq". "units"`