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Three circle with radii 3, 2, 1 touch each other making a gap in the middle, find the area of that gap.

 

 Jan 21, 2021
 #1
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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

 Jan 21, 2021
 #2
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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

 

cool cool cool

 Jan 21, 2021

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