# Circle A has a center at (2 ,8 ) and an area of 8 pi. Circle B has a center at (3 ,2 ) and an area of 27 pi. Do the circles overlap?

Feb 20, 2016

Checking to see if the sum of the radii of the circles is greater than the distance between the circles' centers, we find that yes, they do overlap.

#### Explanation:

The distance between two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ is given by

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

Applying that in this case, we get the distance between the centers of the circle to be

$\sqrt{{\left(3 - 2\right)}^{2} + {\left(2 - 8\right)}^{2}} = \sqrt{1 + 36} = \sqrt{37}$

Then, the circles only overlap if the sum of their radii is greater than $\sqrt{37}$.

The area of a circle with radius $r$ is given by

$\text{area} = \pi {r}^{2}$

Then, letting ${r}_{a}$ be the radius of circle $A$ and ${r}_{b}$ be the radius of circle $B$, we have

$\left\{\begin{matrix}\pi {r}_{a}^{2} = 8 \pi \\ \pi {r}_{b}^{2} = 27 \pi\end{matrix}\right.$

$\implies \left\{\begin{matrix}{r}_{a} = \sqrt{8} \\ {r}_{b} = \sqrt{27}\end{matrix}\right.$

As a matter of estimation, we can tell that

${r}_{a} = \sqrt{8} \approx \sqrt{9} = 3$
$r + b = \sqrt{27} \approx \sqrt{25} = 5$

and

$\sqrt{37} \approx \sqrt{36} = 6$

meaning we should expect ${r}_{a} + {r}_{b} > \sqrt{37}$ and thus for the circles to overlap.

If we actually calculate the values, we get

${r}_{a} + {r}_{b} = \sqrt{8} + \sqrt{27} \approx 8.02458 > 6.08276 \approx \sqrt{37}$

meaning the estimation was correct, and the circles do overlap.