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wu :: forums    riddles    easy (Moderators: ThudnBlunder, Grimbal, towr, Eigenray, Icarus, william wu, SMQ)    Area between 2 circles? « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Area between 2 circles?  (Read 646 times) Worcester Guest Area between 2 circles?   « on: Sep 21st, 2005, 10:49am » Quote Modify Remove 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.     IP Logged Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: Area between 2 circles?   « Reply #1 on: Sep 21st, 2005, 12:30pm » Quote Modify If there is an answer (and I seem to believe there is), it must be d2*pi/4. IP Logged Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: Area between 2 circles?   « Reply #2 on: Sep 21st, 2005, 12:37pm » Quote Modify hidden: 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 IP Logged Neelesh Junior Member     Gender: Posts: 147 Re: Area between 2 circles?   « Reply #3 on: Sep 21st, 2005, 12:40pm » Quote Modify on Sep 21st, 2005, 12:30pm, 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? « Last Edit: Sep 21st, 2005, 12:42pm by Neelesh » IP Logged Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: Area between 2 circles?   « Reply #4 on: Sep 21st, 2005, 1:04pm » Quote Modify on Sep 21st, 2005, 12:40pm, 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... IP Logged Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: Area between 2 circles?   « Reply #5 on: Sep 21st, 2005, 1:45pm » Quote Modify on Sep 21st, 2005, 1:04pm, 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. IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------=> easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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