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        riddles >> easy >> Area between 2 circles?
        
(Message started by: Worcester on Sep 21st, 2005, 10:49am)
Title: Area between 2 circles?
Post by Worcester on Sep 21st, 2005, 10:49am
A coworker gave me this puzzle, claiming it was from a  current GCSE (16 yr olds)  paper. The line d is tangental to the smaller circle. Both circles are around a common point. Can you give the area of the green shading just in terms of d?

He claims there's an answer but I don't think there's enough information.


http://homepage.ntlworld.com/james.holohan/circle.jpg
Title: Re: Area between 2 circles?
Post by Grimbal on Sep 21st, 2005, 12:30pm
If there is an answer (and I seem to believe there is), it must be d2*pi/4.
Title: Re: Area between 2 circles?
Post by Grimbal on Sep 21st, 2005, 12:37pm
[hideb]
If the small radius is r, the large R,
the area is
A = R2*pi - r2*pi
A = (R2 - r2)*pi

If you draw a triangle at the 3 only meaningful points, i.e. the center, the tangent point and the end of the line, you see a right triangle with sides r, d/2 and R.  Pythagoras says:
r2 + (d/2)2 = R2
or
R2 - r2 = (d/2)2

This in the previous formula gives
A = (d/2)2*pi = d2*pi/4
[/hideb]
Title: Re: Area between 2 circles?
Post by Neelesh on Sep 21st, 2005, 12:40pm

on 09/21/05 at 12:30:07, Grimbal wrote:
If there is an answer (and I seem to believe there is), it must be d2*pi/4.


From your statement it appears that you first "predicted" or "imagined" or "visualised" the answer and then gave a formal proof.

If that is so, could you please throw some light on how could you imagine the answer first?
Title: Re: Area between 2 circles?
Post by Aryabhatta on Sep 21st, 2005, 1:04pm

on 09/21/05 at 12:40:36, Neelesh wrote:
From your statement it appears that you first "predicted" or "imagined" or "visualised" the answer and then gave a formal proof.

If that is so, could you please throw some light on how could you imagine the answer first?


My guess about how Grimbal guessed it: I think he imagined the inner circle to be of radius zero.. in which case d is the diameter of the outer circle...
Title: Re: Area between 2 circles?
Post by Grimbal on Sep 21st, 2005, 1:45pm

on 09/21/05 at 13:04:10, Aryabhatta wrote:
My guess about how Grimbal guessed it: I think he imagined the inner circle to be of radius zero.. in which case d is the diameter of the outer circle...

Exactly.  If there is a solution, it must be valid for the simplest case.


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