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avatar+205 

Circles

 Nov 26, 2020
edited by parpar1  Nov 26, 2020
 #1
avatar+116126 
+2

This is an interesting problem......

Let R be the radius of the larger circle  and r  the radius of the smaller circle

 

Note  that  F = (0, R)   and   H = (R,0)

 

FH =   sqrt ( R^2 + R^2 )  =  R*sqrt (2)

Likewise BC = r*sqrt (2)

 

Midpoint of  FH  =  (R/2  R/2)

Distance  from O  to the midpoint =  R/sqrt (2)

Midpoint of BC =   (r/2, r,2) 

Distance  from O to this midpoint  = r/sqrt (2)

 

The  quadrilateral is a  trapezoid  with  height =  (R + r) / sqrt (2)

And  the sum of its bases =  sqrt (2) (R + r)

 

Area of  quadrilateral =   (1/2) (R + r)  / sqrt (2)  *  ( sqrt (2) (R + r) )  =  (1/2) ( R + r)^2

 

Area  between the circles  =  pi (R^2  - r^2)

 

Area of quadrilateral / area between circles =   2/3

 

So

 

(1/2) (R + r)^2             2

___________  =       ____

pi (R^2 - r^2)               3

 

 

(1/2) (R + r) (R + r)           2

_______________  =      ___

pi ( R + r) ( R - r)               3

 

 

(1/2) (R + r)                2

_________  =         ________      cross-multiply

pi (R - r)                      3

 

(3/2) ( R + r)  = 2pi ( R - r)

 

(3/2)R  + (3/2)r  =  2pi R  - 2pi r

 

( 2pi + 3/2  ) r =  (2pi - 3/2) R

 

r / R  =  (2pi - 3/2)  / (  2pi + 3/2 ) ≈  .6145

 

 

cool cool cool

 Nov 26, 2020
 #2
avatar+1039 
+2

Large circle radius R = ?        Small circle radius r = 1

 

Large circle area            Al = R2pi           Small circle area      As = pi

 

Trapezoid area               At = R2/2 + R + 1/2

 

Annular ring area           Ar = R2pi - pi

 

R2pi - pi  / (R2/2 + R + 1/2) = 3/2

 

R = 1.62719711 units

 

So, the ratio is      1 : 1.62719711       or        0.614553697

 

 Nov 26, 2020
edited by jugoslav  Nov 26, 2020

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