Question

Which is larger, the light area or the dark area? Please share your working also?

Answer 1

Both arear of light ,and dark zone would be equal.

here is the explanation:
let the radius of small circle is=r
area of circle would be =
the total area of the four lens-shaped regions formed by the pairwise intersection of circles is:
follow this link to find out how: http://mathworld.wolfram.com/Circle-CircleIntersection.html
now, area of all four light zone=area of bigger circle-area covered by all four small circle
area of bigger circle=4 as radius of bigger circle is 2r
area covered by all four small circle B=4 -A
hence area of all four light zone=4 -B =A
now area of one light zone is=A/4 ,and area of one lens shaped part is A/4 (both are equal).

Answer 2

Let the radius of the inner circles be : r
Let us calculate the area of the dark zone, as follows:
Dark area = Area of square of side r - 2 times (area of square of side r - 1/ 4 area of inner circle)
Dark area = r exp 2 - 2 ( r exp 2 - 3.1416 * r exp 2 / 4 ) = r exp 2 * ( 3.1416 / 2 - 1)
Light area = 1/4 area of big circle - 1/2 area of inner circle - 1/2 area of inner circle + area of dark zone (intersection of two circles)
Light area = 1/4 * 3,1416 + (2 r) exp 2 - 3,1416 / 2 * r exp 2 - 3,1416 / 2 * r exp 2 + r exp 2 * (3,1416 / 2 - 1 ) = r exp 2 * (3,1416 / 2 -1)
Therefore, both areas are equal