Circle A has a center at (-2 ,-1 ) and a radius of 3 . Circle B has a center at (-8 ,3 ) and a radius of 2 . Do the circles overlap? If not what is the smallest distance between them?

Jun 19, 2016

They do not overlap and the distance is $2.21$.

Explanation:

Two circles overlaps if the distance between the two centers is smaller or equal to the sum of the radius. It is quite clear looking at the picture. We can calculate the distance between the centers:

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

$= \sqrt{{6}^{2} + {\left(- 4\right)}^{2}} = \sqrt{36 + 16} = \sqrt{52} \setminus \approx 7.21$.

The sum of the two radius is ${r}_{A} + {r}_{B} = 3 + 2 = 5$, then the two circles will not overlap.
The distance between them is the difference between the distance of the centers and the sum of the two radius:

$7.21 - 5 = 2.21$.