# Circle A has a center at (1 ,5 ) and an area of 12 pi. Circle B has a center at (8 ,1 ) and an area of 3 pi. Do the circles overlap? If not, what is the shortest distance between them?

##### 1 Answer
Dec 27, 2016

We first get the radii of the two circles.

#### Explanation:

Since $A = \pi {r}^{2} \to r = \sqrt{\frac{A}{\pi}}$
Circle A: ${r}_{A} = \sqrt{\frac{12 \cancel{\pi}}{\cancel{\pi}}} = \sqrt{12} = 2 \sqrt{3}$
Circle B: ${r}_{B} = \sqrt{\frac{3 \cancel{\pi}}{\cancel{\pi}}} = \sqrt{3}$

Then the distance between the centers:

${D}^{2} = {\left(\Delta x\right)}^{2} + {\left(\Delta y\right)}^{2}$ (Pythagoras)
${D}^{2} = {\left(8 - 1\right)}^{2} + {\left(1 - 5\right)}^{2} = {7}^{2} + {4}^{2} = 49 + 16 = 65$
$\to D = \sqrt{65} \approx 8.06$

Together:
The radii add up to $2 \sqrt{3} + \sqrt{3} = 3 \sqrt{3} = \sqrt{27} \approx 5.20$

This is much smaller than the distance, so they do not overlap. The smallest distance between them is $\sqrt{65} - \sqrt{27} \approx 2.87$