Answer:If an N-gon (polygon with N sides) has perimeter P, then each of the
N sides has length P/N. If we connect two adjacent vertices to the
center, the angle between these two lines is 360/N degrees, or 2*pi/N
radians. (Do you understand radian measure well enough to follow me
this far?)
The two lines I just drew, plus the side of the polygon between them,
form an isosceles triangle. Adding the altitude of the isosceles
triangle makes two right triangles, and we can use one of them to
derive the equation
sin(theta/2) = s/(2R)
where theta is the apex angle (which I said is 2*pi/N radians), R is
the length of the lines to the center (the radius of the circumscribed
circle), and s is the length of the side (which I said is P/N).
Putting those values into the equation, we have
sin(pi/N) = P/(2NR)
so that
P = 2NR sin(pi/N)
gives the perimeter of the N-gon with circumradius R.
Can we see a connection between this formula and the perimeter of a
circle? The perimeter of the circumcircle is 2*pi*R. As we increase
N, the perimeter of the polygon should get closer and closer to this
value. Comparing the two, we see
2NR*sin(pi/N) approaches 2*pi*R
N*sin(pi/N) approaches pi
You can check this out with a calculator, using big numbers for N such
as your teacher's N=1000. If you calculate the sine of an angle in
degrees rather than radians, the formula will look like
N*sin(180/N) --> pi
Step-by-step explanation:
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