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Perimeter of an Inscribed Regular Polygon


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.  

Thank you for your time,
Aaron Willems


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

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