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 )!