# Circle A has a center at (2 ,4 ) and an area of 81 pi. Circle B has a center at (4 ,3 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Nov 7, 2016

The larger circle completely encloses the smaller circle.

#### Explanation:

Here is a graph of the two circles: Nov 8, 2016

Smaller circle lies within bigger circle. Smallest distance between them is $0.764$

#### Explanation:

Circle A has a center at $\left(2 , 4\right)$ and its radius is $\sqrt{\frac{81 \pi}{\pi}} = 9$

Circle B has a center at $\left(4 , 3\right)$ and its radius is $\sqrt{\frac{36 \pi}{\pi}} = 6$

The distance between centers is

$\sqrt{{\left(4 - 2\right)}^{2} + {\left(3 - 4\right)}^{2}} = \sqrt{{2}^{2} + {1}^{2}} = \sqrt{5}$

As the distance at $\sqrt{5}$ between them is less than difference in radius which is $9 - 6 = 3$

smaller circle lies within bigger circle. For details see here

Also see graph below (the other answer).

Smallest distance between them is $9 - 6 - \sqrt{5} = 0.764$