Question

Circle A has a center at `(5 ,4 )` and an area of `54 pi`. Circle B has a center at `(12 ,8 )` and an area of `25 pi`. Do the circles overlap?

Answer 1

Two circles overlap (intersect) each other.

Explanation:

Circle A has a center at `(5,4)` and its radius is `sqrt((54pi)/pi)=sqrt54=7.348`

Circle B has a center at `(12,8)` and its radius is `sqrt((25pi)/pi)=5`

Sum of the radii is `12.348` and difference is `2.348`

The distance between centers is

`sqrt((12-5)^2+(8-4)^2)=sqrt(7^2+4^2)=sqrt65=8.062`

The distance between centers at `8.062` is less than sum of radii and greater than difference in radii.

Hence, two circles intersect each other. For details see here
graph{(x^2+y^2-10x-8y-13)(x^2+y^2-24x-16y+183)=0 [-13.92, 26.08, -4.96, 15.04]}