# Circle A has a center at (5 ,2 ) and an area of 18 pi. Circle B has a center at (3 ,6 ) and an area of 27 pi. Do the circles overlap?

Feb 19, 2016

The circles do overlap (by $4.87$ units)

#### Explanation:

Since Circle A has an Area $= 18 \pi$,
Circle A has a Radius of ${r}_{A} = \sqrt{18} = 3 \sqrt{2} \approx 4.24$
$\textcolor{w h i t e}{\text{XXX}}$(this follows from formua $\text{Area} = \pi {r}^{2}$)

Since Circle B has an Area $= 27 \pi$
Circle B has a Radius of ${r}_{B} = \sqrt{27} = 3 \sqrt{3} \approx 5.20$

The distance between the center of A at $\left(5 , 2\right)$ and the center of B at $\left(3 , 6\right)$ is
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(5 - 3\right)}^{2} + {\left(2 - 6\right)}^{2}} = 2 \sqrt{5} \approx 4.47$

Together the radii of circles A and B cover $4.24 + 5.20 = 9.34$ of the line segment joining the centers of A and B.
Since this is greater than the actual length of the line segment joining A and B, the radii must overlap by $9.34 - 4.47 = 4.87$ units.
graph{((x-5)^2+(y-2)^2-18)((x-3)^2+(y-6)^2-27)=0 [-14.11, 17.94, -3.74, 12.28]}