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?

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Answer 1 · 2016-10-06T03:52:06

The circles overlap

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 $sqrt(12) + sqrt(24) ~~ 8.36$

$d = sqrt((5 - 3)² + (2 - 1)²)$

$d = sqrt((4)² + (1)²)$

$d = sqrt(5)$

$d ~~ 2.23$

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.

Circle A has a center at (3 ,1 )(3 ,1 ) and an area of 12 pi12 pi. Circle B has a center at (5 ,2 )(5 ,2 ) and an area of 24 pi24 pi. Do the circles overlap?