Perimeter of an Inscribed Regular PolygonDate: 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/ |
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