SOLUTION: In the figure below, the innermost circle has radius 1.
It is circumscribed by an equilateral triangle, which is circumscribed by a circle, which is circumscribed by a square,
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-> SOLUTION: In the figure below, the innermost circle has radius 1.
It is circumscribed by an equilateral triangle, which is circumscribed by a circle, which is circumscribed by a square,
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Question 1031251: In the figure below, the innermost circle has radius 1.
It is circumscribed by an equilateral triangle, which is circumscribed by a circle, which is circumscribed by a square, which is circumscribed by yet another circle, and so forth.
Answer two questions:
1. What is the radius of the outermost circle?
2. Is finding the value of this radius an exercise/problem in discrete mathematics or continuous mathematics?
Thank you! Found 2 solutions by rothauserc, ikleyn:Answer by rothauserc(4718) (Show Source):
You can put this solution on YOUR website! This is a problem in discrete mathematics, consider the definition of a circle in discrete mathematics
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the circle is comprised of polygons of n sides as n approaches infinity
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begin with the circle of radius 1 inscribed in the equilateral triangle
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the radius of the next circle is 1 / sin 30 = 2
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the interior angle of regular polygon is (n-2)180/n
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the hypotenuse (radius of the next circle) = radius of previous circle / the sin of (interior angle / 2)
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the radius of the next circle is 2 / sin 45 = 2.828427125
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the radius of the next circle is 2.828427125 / sin 54 = 3.496128196
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the radius of the next circle is 3.496128196 / sin 60 = 4.03698111
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the radius of the next circle is 4.03698111 / sin 64.28 = 4.480926229
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the radius of the next circle is 4.480926229 / sin 67.5 = 4.8501196
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the radius of the next circle is 4.8501196 / sin 70 = 5.161389472
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now we look for the limit of this sequence
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The limit of this sequence is "Polygon Circumscribing Constant" which is defined as
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limit for n > = 3 of the product of 1 / cos(pi/n) = 8.7000366
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