Question

Given that the radius of the circumcircle (large circle) is `r`, evaluate the ratio of the area of A regions to B regions - `(sumA_i)/(sumB_i)`?

Answer 1

Radius of the large circumcircle`=r`

Area of this large circle `=pir^2`

The length of each side of the square at the center is equal to the radius of the large circumcircle.

So area of the square `r^2`

Radius of each of four small circle`=r/2`

So its area `=pi(r/2)^2=1/4pir^2`

From the figure it is obvious that

`sumA_i="area of large circle"-"area of square"-2xx"area of small circle"`

`=pir^2-r^2-2xx1/4pir^2=(pi/2-1)r^2`

Each of the petal of flower at the center is composed of two similar segments of small circle and 8 such segments form the complete flower.

Area of each such segment
`=1/4(1/4pir^2)-1/2(r/2)^2`

`=(1/16pi-1/8)r^2`

So `sumB_i=8xx(1/16pi-1/8)r^2`

`=(pi/2-1)r^2`

So the asked ratio

`(sumA_i)/(sumB_i)=((pi/2-1)r^2)/((pi/2-1)r^2)=1/1`