Circles around a Larger CircleDate: 07/26/2003 at 10:38:44 From: Kelly Jones Subject: Circles Is there a formula to determine the diameter of several smaller circles outlining the circumference of a larger circle? For example, if I have a circle with a circumference equalling 10 and I want to make 10 smaller circles fit perfectly around this circumference, what diameter would these smaller circles have? And what if I only wanted to make 8 small circles fit, or any other number? Date: 07/27/2003 at 17:44:24 From: Doctor Jeremiah Subject: Re: Circles Hi Kelly, Lets say you want n circles of radius r around the outside of a middle circle of radius R. If you draw a line from the center of the middle circle that makes a tangent with the outside circle, you end up with this triangle: +++++++ ++ ++ + + + + \ A \ + + + --- \ + + | + | \+ r | + | \ + r | + | + | B \+ | + | \ ++ | ++ r+R \ +++++++ | +++++ \ | +++++ | +++ \ | | +++ \ | | \ R | \ | | \ | | \a| | \| | + --- C a = (360/n)/2 because if it were the full angle between two tangent lines it would be 360/n. The tangent line makes a 90-degree angle with the radius coming from the center of the outside circle. So we have a right triangle that looks like this (just rotated a bit): C + /| /a| / | / | / | / | R+r | / | / | / | / 90| +-----r-----+ A B Where, again, a = (360/n)/2 which, if we simplify it a bit, is the same as a = 180/n Using trigonometry: sin(180/n) = r/(R+r) (R+r) sin(180/n) = r R sin(180/n) + r sin(180/n) = r R sin(180/n) = r - r sin(180/n) R sin(180/n) = r(1 - sin(180/n)) R sin(180/n)/(1 - sin(180/n)) = r So, if you know R (the radius of the middle circle) and n (the number of circles around the outside) you can calculate the radius of the outside circle. For example: If you want n=10 outside circles and the middle circle has a circumference of 10, then its radius is R=10/(2 Pi) and you can do this: R sin(180/n)/(1 - sin(180/n)) = r 10/(2 Pi) sin(180/10)/(1 - sin(180/10)) = r 5/Pi sin(18)/(1 - sin(18)) = r r = 0.71 So the outside circles would have to have a diameter of 2*.71 = 1.42 - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ |
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