Circle A has a center at (2 ,2 ) and an area of 8 pi. Circle B has a center at (13 ,6 ) and an area of 54 pi. Do the circles overlap?

May 29, 2016

No, they don't overlap.

Explanation:

I start calculating the radius of the two circles.
We know that $A = \pi {r}^{2}$ then $r = \sqrt{\frac{A}{\pi}}$
The radius of the circles, that I will call respectively ${r}_{1}$ and ${r}_{2}$ are

${r}_{1} = \sqrt{8 \frac{\pi}{\pi}} = \sqrt{8} \setminus \approx 2.83$
${r}_{2} = \sqrt{54 \frac{\pi}{\pi}} = \sqrt{54} \setminus \approx 7.35$.

If the distance between the two centers is bigger than the sum of the two radius, the two circles will not overlap, otherwise they will overlap.

The distance between the two centers is

$d = \sqrt{{\left(2 - 13\right)}^{2} + {\left(2 - 6\right)}^{2}} = \sqrt{{\left(- 11\right)}^{2} + {\left(- 4\right)}^{2}}$
$= \sqrt{137} \setminus \approx 11.7$.

The sum of the two radius is $2.83 + 7.35 = 10.18$ and $11.7 > 10.18$ so the circles cannot overlap.