bonanova Posted August 21, 2009 Report Share Posted August 21, 2009 Sketch four circles, two side by side and two more directly above them. All four circles have the same diameter and touch their two nearest neighbors. Quickly determine the area of the space between the circles. Extra credit: Sketch three circles, two side by side and a third resting above and between them. All three circles have the same diameter and touch the other two. Quickly determine the area of the space between the circles. Enjoy, and please use spoilers. Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted August 21, 2009 Report Share Posted August 21, 2009 Reveal hidden contents R2(4-pi) (square whatever unit of measure) and Reveal hidden contents r(3).5-1/2piR2 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted August 21, 2009 Author Report Share Posted August 21, 2009 On 8/21/2009 at 12:05 PM, plainglazed said: Reveal hidden contents R2(4-pi) (square whatever unit of measure) and Reveal hidden contents r(3).5-1/2piR2 Yes. There is a cute way to arrive at the answer without performing a calculation and to phrase it without using numbers or symbols. Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted August 21, 2009 Report Share Posted August 21, 2009 (edited) For the first part Reveal hidden contents the area is that of a square minus the largest possible inscribed circle For the second part Reveal hidden contents the area of the equilateral triangle of leg twice the radius of a circle less half the area of the circle Edit: added second part Edited August 21, 2009 by plainglazed Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted August 21, 2009 Author Report Share Posted August 21, 2009 On 8/21/2009 at 12:25 PM, plainglazed said: For the first part Reveal hidden contents the area is that of a square minus the largest possible inscribed circle For the second part Reveal hidden contents the area of the equilateral triangle of leg twice the radius of a circle less half the area of the circle Edit: added second part Reveal hidden contents The shapes you mention are formed by connecting the centers of the circles. And constructing them verifies the results without calculation. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 24, 2009 Report Share Posted August 24, 2009 Reveal hidden contents is it directly related to the size of the circles...e.g. its an eighth of one circle Sorry if this is completely wrong. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 24, 2009 Report Share Posted August 24, 2009 On 8/21/2009 at 12:05 PM, plainglazed said: Reveal hidden contents R2(4-pi) (square whatever unit of measure) and Reveal hidden contents r(3).5-1/2piR2 Reveal hidden contents I believe the second part is R2(3).5-1/2piR2, right? Quote Link to comment Share on other sites More sharing options...
Question
bonanova
Sketch four circles, two side by side and two more directly above them.
All four circles have the same diameter and touch their two nearest neighbors.
Quickly determine the area of the space between the circles.
Extra credit:
Sketch three circles, two side by side and a third resting above and between them.
All three circles have the same diameter and touch the other two.
Quickly determine the area of the space between the circles.
Enjoy, and please use spoilers.
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