Equilateral Shapes Inscribed in a CircleDate: 04/07/2003 at 15:02:43 From: Ashwini Prasad Subject: Inscribed equilateral shapes in a circle Is there a general formula for the length of a side of an equilateral shape that is inscribed in a circle? Date: 04/07/2003 at 19:26:57 From: Doctor Dotty Subject: Re: Inscribed equilateral shapes in a circle Hi Ashwini, Thanks for the question. Let n denote the number of sides on the polygon. Here is a square (n = 4) inscribed in a circle (radius r). A triangle has been drawn connecting two of the vertices of the square and the centre of the circle. * * *_ _ _ _ _ _*_ |. | * | . r |* | . | * | x( . | * | . | * | . r |* |. _ _ _ _ _ _ | * * * * There are 360 degrees in a complete circle. You could fit one of these triangles around the centre for each side of the polygon, so angle x is one quarter of 360, which is 90 degrees. Generally, angle x would equal 360/n. The cosine rule states that on a triangle: /\ /C \ / \ b/ \a / \ / \ / \ /)A_ _ _ _ _ B(\ c a^2 = b^2 + c^2 - 2*b*c*Cos(A) [If you're not familiar with the cosine rule, write back and I'll explain - it looks a lot more complicated than it is.] On our triangle, 'b' and 'c' are equal to r. So: a^2 = r^2 + r^2 - 2*r*r*Cos(360/n) a^2 = 2r^2 - 2(r^2)Cos(360/n) a^2 = 2(r^2)(1 - Cos(360/n)) a = sqrt[2(r^2)(1 - Cos(360/n))] Where 'a' is the length of a side, 'r' is the radius of the circle, and 'n' is the number of sides on the polygon. This is true when n is an integer greater than 2. Does that help? If I can help any more with this problem or any other, please write back. - Doctor Dotty, The Math Forum http://mathforum.org/dr.math/ Date: 04/08/2003 at 12:46:04 From: Ashwini Prasad Subject: Thank you (Inscribed equilateral shapes in a circle) Doctor Dotty, I wrote to you yesterday about the length of the side of a figure inscribed in a circle. I wanted to thank you so much for the thorough explanation. Also, thanks for such a prompt reply. Ashwini |
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