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Random math patterns time. If you take a quarter, or any other coin, you can fit 6 other coins around it so they are all touching the center coin and their neighbors on either side. Why 6? I dunno. But what size does the middle one have to be for 5 outside circles? or 4?

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Start

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So lets assume the radius of the circles on the outside (our quarters) is 1. A quarter is 2 quarter-radius' in diameter...go figure.

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If you draw a line between the centers of our circles you will get a bunch of angles. In the case of our 6 quarters, we get a hexagon."; echo "
The sum of the outer angles for a polygon is ∑a = 180*(n-2), so for a hexagon we get 720 degrees, or 120 degrees for each angle."; echo "
Now if you draw a 2 lines from the center of 2 next door circles into the center you get a triangle. Each of the 2 outer angles will be half the original outer angle (120/2 = 60) and the center will be 180 minus the sum of the 2 angles (180-60-60=60)."; echo "
So, how can we compare the number of outer circles to the inner angles?

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inner angle = 180 - (2 * ( ∑a / 2 ) / x)"; echo "
or simplified:   inner angle = 180 - ( ∑a / x )

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So then we can calculate out for each shape:

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# of Outer CirclesInternal Angle
3120
490
572
660
845
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Inner Circle Diameter

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Now, how do we relate the inner angle to the radius of the middle circle?"; echo "
Something like this:

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inner radius = outer radius / ( SIN( inner angle / 2 ) - outer radius"; echo "
with numbers:   inner radius = 1 / ( SIN( inner angle / 2 ) - 1

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So:

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# of Outer CirclesInternal AngleInternal RadiusInternal Diameter
31200.1550.309
4900.4140.828
5720.7011.403
6601.0002.000
8451.6132.610
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And This Affects Me How?

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Hopefully not at all really. But according to wikipedia and some quicky ratio-ization lets have some fun:

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CoinDiameterCircle Design
Penny19.056 outer : 1 inner
Nickle21.206 outer : 1 inner
Dime18.036 outer : 1 inner
Quarter23.886 outer : 1 inner
50 cent27.136 outer : 1 inner
5 outer : 1 inner penny
Loonie26.506 outer : 1 inner
5 outer : 1 inner dime (close)
Toonie28.006 outer : 1 inner
5 outer : 1 inner penny (close)
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Well that was really an excersize in letdown...oh well, my questions were answered, and your time wasted, so 2 points to me!

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