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Equilateral Shapes Inscribed in a Circle

Date: 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
Associated Topics:
High School Conic Sections/Circles
High School Triangles and Other Polygons

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