# 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?

Nov 8, 2016

Two circles overlap (intersect) each other.

#### Explanation:

Circle A has a center at $\left(5 , 4\right)$ and its radius is $\sqrt{\frac{54 \pi}{\pi}} = \sqrt{54} = 7.348$

Circle B has a center at $\left(12 , 8\right)$ and its radius is $\sqrt{\frac{25 \pi}{\pi}} = 5$

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

The distance between centers is

$\sqrt{{\left(12 - 5\right)}^{2} + {\left(8 - 4\right)}^{2}} = \sqrt{{7}^{2} + {4}^{2}} = \sqrt{65} = 8.062$

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]}