# Circle A has a center at (1 ,5 ) and an area of 12 pi. Circle B has a center at (7 ,2 ) and an area of 8 pi. Do the circles overlap? If not, what is the shortest distance between them?

Feb 16, 2016

We can find the radii of the circles from their areas. If the distance between their centers is less than the sum of the two radii, the circles overlap. If not (as in this case), the distance between the centers minus the sum of the radii equals the distance between the closest points: $6.71 - 6.29 = 0.42$.

#### Explanation:

We can find the radius of a circle from its area:

$A = \pi {r}^{2}$

Rearranging:

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

The radius of the circle with area $12 \pi$ will be $\sqrt{12} \approx 3.46$ units. The radius of the circle with area $8 \pi$ will be $\sqrt{8} \approx 2.83$.

The distance between the centers of the two circles is given by:

$l = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}} = \sqrt{{\left(7 - 1\right)}^{2} + {\left(2 - 5\right)}^{2}} = \sqrt{45} \approx 6.71$

The sum of the two radii is $3.46 + 2.83 = 6.29$. This is less than the distance between the two centers, $6.71$, which means that the two circles do not overlap.

The distance between their closest points is $6.71 - 6.29 = 0.42$ units.