# Circle A has a center at (12 ,2 ) and an area of 13 pi. Circle B has a center at (3 ,6 ) and an area of 28 pi. Do the circles overlap?

##### 1 Answer
Feb 18, 2016

We can find the radius of each circle given its area. If the centers are further apart than the sum of the two radii, the circles don't overlap. If not, they do. The radii are 3.6 and 5.3 units, totaling 8.9 units, and the distance between the centers is 9.85 units, so the circles do not overlap.

#### Explanation:

The area of a circle is given by $A = \pi {r}^{2}$. Rearranging to make $r$ the subject:

$r = \sqrt{\frac{A}{\pi}}$

Circle A:

$r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{13 \pi}{\pi}} = \sqrt{13} \approx 3.6$

Circle B:

$r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{28 \pi}{\pi}} = \sqrt{28} \approx 5.3$

The sum of these two radii is $8.9$ units.

The distance between the centers is given by:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$= \sqrt{{\left(12 - 3\right)}^{2} + {\left(6 - 2\right)}^{2}} = \sqrt{81 + 16} = \sqrt{97} \approx 9.85$ units.

Since this is greater than the sum of the radii, the circles do not overlap.