# Circle A has a center at (2 ,2 ) and an area of 18 pi. Circle B has a center at (13 ,6 ) and an area of 27 pi. Do the circles overlap?

Oct 27, 2016

There is no Overlap

#### Explanation:

so we have two circles,

A, with centre $\left(2 , 2\right)$ and Area $18 \pi$
B, with centre $\left(13 , 6\right)$ and Area $27 \pi$

We will work with A first

$A = \pi {r}^{2}$

$18 \pi = \pi {r}^{2}$

$r = \sqrt{18} = 3 \cdot \sqrt{2}$

Now B,

$27 \pi = \pi {r}^{2}$

$r = \sqrt{27} = 3 \cdot \sqrt{3}$

so if they overlap the distance between the centres of the circles will be less than the two radii.

Distance between the two circles,

$\vec{A B}$=$B - A$

$= \left(13 , 6\right) - \left(2 , 2\right)$

$= \left(11 , 4\right)$

Using Pythagoras theorem,

Distance = $\sqrt{{a}^{2} + {b}^{2}}$

$= \sqrt{{11}^{2} + {4}^{2}}$

$= \sqrt{137}$

Now to find out,
To overlap this must be true,

$\sqrt{137} < {r}_{\text{A"+r_"B}}$

$\sqrt{137} < 3 \cdot \sqrt{2} + 3 \cdot \sqrt{3}$

$11.7047 < 4.2426 + 5.1962$

$11.7047 < 9.4388$

This is not true so there is no Overlap.

Visually,

The line f is longer than the two radii combined.