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### Perimeter of an Inscribed Regular Polygon

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Date: 12/10/98 at 09:17:06
From: Aaron Willems
Subject: Polygons and the perimeter of a polygon

I am trying to figure out how to find the perimeter of a polygon, and
one that is inscribed in a circle.

Can you give me the formulas for the perimeter of an inscribed polygon?
And the formula for the area?

If you could help, I would really appreciate it.

Aaron Willems
```

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Date: 12/10/98 at 12:36:40
From: Doctor Wilkinson
Subject: Re: Polygons and the perimeter of a polygon

I assume you are talking about a regular polygon, one with all sides
and angles equal.

Suppose r is the radius of the circle and the polygon has n sides. Draw
line segments connecting the center of the circle and the vertices of
the polygon. This divides the circle into n triangles. The triangles
are all isosceles triangles with two sides equal to r and an unknown
(so far) base. If we could figure out the base of each triangle, then
we could just multiply by n to get the circumference, and we could also
figure out the area of each triangle and multiply by n to get the area
of the polygon.

We can find the base by trigonometry if we can figure out the angle
opposite. But now the angles opposite the bases add up to 360 degrees,
and there are n of them, so each angle is 360/n. If you draw a
perpendicular from the center of the circle to the base of one of the
triangles you divide it into two right triangles, with the angle oppose
half the base being half of our angle of 360/n, or 180/n. The
hypotenuse of each right triangle is r, so half the base is r sin
(180/n) and the base is 2r sin(180/n). Now you should be able to
finish.

- Doctor Wilkinson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons
High School Trigonometry

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