Circle A has a center at #(1 ,3 )# and an area of #16 pi#. Circle B has a center at #(2 ,7 )# and an area of #28 pi#. Do the circles overlap?

1 Answer
Feb 19, 2016

The distance from the centre of circle A to the centre of circle B is less than the sum of the two radii. Thus we can conclude that the two circles are overlapping.

Explanation:

area of a circle = # pi r^2#
area of circle A #= 16 pi#
area of circle B #= 28 pi#
radius of circle A = #r_A = sqrt((16pi)/pi) = sqrt(16) = 4#
radius of circle B #= r_B = sqrt((28pi)/pi) = sqrt(28) = 2sqrt(7)#

If the circles are just touching, the distance between their centres will be:
#r_A+r_B = 4 + 2sqrt(7) ~= 9.29#

The distance from the centre of circle A to the centre of circle B = d
Using Pythagorus's theorem:
#d^2 = (1-2)^2 + (3-7)^2#
#d^2 = (-1)^2 + (-4)^2#
#d^2 = (1) + (16) = 17#
#d = sqrt(17) ~= 4.12#

d is less than the sum of the two radii. Thus we can conclude that the two circles are overlapping.

They actually overlap a lot as the centre of circle A is within circle B (because #r_B > d# )!