Three circle with radii 3, 2, 1 touch each other making a gap in the middle, find the area of that gap.
h=52−32⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√=25−9⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√=16⎯⎯⎯⎯√=4.h=52−32=25−9=16=4.
so the area of the triangle is 12×6×4=12in.2. there is a triangle in the middle
+128407 If we join the centers of each circle, we will form a 3-4-5 right triangle
The area of this triangle is (3*4)/2 = 6
The angle formed by the lines joining the two lager circles to the center of the smallest circle = 90°
So the area of the sector of 1/4 of the smallest circle with a central measure of 90° is pi (1)^2 *(90/360) = pi/4
The angle (theta) formed by joining the center of the larger circle with the two smaller circles can be forund as
arcsin (3/5) ≈ 36.87°
So the area of a sector in the larger circle with a central measure of 36.87° is given by
pi ( 3)^2 (36.87/360) ≈ .9271 pi
And in the circle with a radius of 2 we are looking for the area of a sector with a central angle measure of (90-36.87) = 53.13° = pi (2^2) ( 53.13 / 360) ≈ .5903 pi
So.....the area of the gap ≈ 6 - pi ( 1/4 + .9271 + .5903) ≈ .448 units^2
