Question 904401: If a regular polygon of 24 sides is inscribed in a circle of radius 1 cm , then find the length of each side of the polygon
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula is s = 2*r*sin(180/n)
s = length of a side
r = radius of the circle
n = number of sides of the polygon
in your problem this becomes:
s = 2*1*sin(180/24) which becomes s = 2*1*sin(7.5) which becomes s = .2610523844.
the formula is derived from the fact that a regular polygon of 24 sides forms 24 isosceles triangle with side legs of 1 inch each and a central angle, or vertex angle of the isosceles triangle, of 15 degrees.
drop an altitude from the vertex to the base of the isosceles triangle and that forms two right triangles that are congruent to each other.
each of these right triangles has a central angle of 7.5 degrees.
the sine of the central angle of 7.5 degrees is equal to the side opposite the central angle divided by the hypotenuse which give you the formula of sine(7.5) = x/1
solve for x and you get x = 1 * sin(7.5)
you want 2x because x is only 1/2 the length of a side of the polygon, so the formula becomes s = 2*sin(7.5)
that gets you s = .26105.....
2*sin(7.5) is equivalent to the generic formula of 2*r*sin(180/n) where r is equal to 1 and n is equal to 24.
the formula becomes s = 2*1*sin(180/24) which becomes s = 2*sin(7.5) as we saw above.
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