comparing areas of unstructured shapes

Question

I want to know, which area is greater? A or B?

I know counting squares is an estimation for areas, but I want to know is there any other way to solve this problem?

Answer 1

Let C be the area of the dark blue region separating A and B. Then A+B+C is a 10 by 10 square, plus two semicircles of diameter 10: $$ A+B+C = 100 + 25\pi $$ B+C is an 8 by 8 square, plus two semicircles of diameter 8: $$ B+C = 64 + 16 \pi $$ B is a 6 by 6 square, plus two semicircles of diameter 6: $$ B = 36 + 9\pi $$ Then going up the chain, $$ C = 28 + 7 \pi $$ $$ A = 36+ 9 \pi $$ A and B have exactly the same area.

Answer 2

The three "hearts" are similar and the ratio of their lengths is $6:8:10=3:4:5$. So, the ratio of their areas is $3^2:4^2:5^2=9:16:25$. Since $25-16=9$, $A$ and $B$ have the same area.