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.