# Circle A has a center at (3 ,1 ) and an area of 12 pi. Circle B has a center at (5 ,2 ) and an area of 24 pi. Do the circles overlap?

Oct 6, 2016

The circles overlap

#### Explanation:

We can use the areas of the circles to compute their respective radii:

A_A = pi(r_A)²
A_B = pi(r_B)²

12pi = pi(r_A)²
24pi = pi(r_B)²

${r}_{A} = \sqrt{12}$
${r}_{B} = \sqrt{24}$

The circles will not overlap if the distance, d, between their centers is greater than $\sqrt{12} + \sqrt{24} \approx 8.36$

d = sqrt((5 - 3)² + (2 - 1)²)

d = sqrt((4)² + (1)²)

$d = \sqrt{5}$

$d \approx 2.23$

The larger circle encloses the smaller if:

$d < \sqrt{24} - \sqrt{12}$

$d < 1.44$

$1.44 < 2.23 < 8.36$, therefore, the circles overlap.