SOLUTION: For this exercise, consider a circle of radius 1, and corresponding inscribed and circumscribed polygons with the number of sides n = 3, 4, 5, 6, and 8. For each n = 3, 4, 5, 6 &

Question 739054: For this exercise, consider a circle of radius 1, and corresponding inscribed and circumscribed polygons with the number of sides n = 3, 4, 5, 6, and 8.
For each n = 3, 4, 5, 6 & 8, what are the perimeters of the inscribed and circumscribed polygons with n sides?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
NOTE:
I do not know what is expected of you because I do not know if this is homework for fifth grade, for college, or somewhere in between. I assume that using trigonometric functions is acceptable. I suspect that the intention is showing (by examples) how the perimeter approaches the circumference of a circle as the number of sides increases. (Note how is always in between the perimeter for the inscribed and circumscribed polygons, an how both approach as the number of sides increases)

THE SOLUTION:
We can only solve the problem if those are regular (symmetrical) polygons, because otherwise the perimeter would change with the shape.
Connecting the vertices to the center, we can split a polygon with sides into congruent isosceles triangles. If we draw the altitudes, we split each isosceles triangle into two congruent right triangles, for a total of right triangles.

is the center of the circle and the polygon; is the side of the polygon
The angles AOP and POB measure (or if you prefer degrees rather than radians)

FOR AN INSCRIBED POLYGON:
A and B are points on the circle and OA and OB are radii with length

The length of the side of the polygon is

and the perimeter is

FOR A CIRCUMSCRIBED POLYGON:
P is on the circle, OP is a radius with lenght
The length of the side of the polygon is

and the perimeter is

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