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### Equilateral Shapes Inscribed in a Circle

```Date: 04/07/2003 at 15:02:43
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
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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