Question

If three circles of radius a each, are drawn such that each touches the other two, prove that the area included between them is equal to `4/25"a"^2`.

Answer 1

When three circles touch each other, their centres form an equilateral triangle, with each side being 2a.

Area of the triangle`=sqrt(3)/4xx2"a"xx2"a" = sqrt(3)"a"^2`

Total area of the three sectors of circles `=3xx60/360xx22/7xx"a"^2 = 1/2xx22/7 "a"^2 = 11/7"a" ^2`

Area of the region between the circles = Area of the triangle - Area of three sectors

`=(sqrt(3)-11/7)"a"^2`

= (1.73 - 1.57)a²

= 0.16 a²

=  0.16 a²

`=4/25"a"^2 `