SOLUTION: a regular polygon has 10 sides.the area is 200 cm^2 find the length of radius and length of side?

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Question 768144: a regular polygon has 10 sides.the area is 200 cm^2 find the length of radius and length of side?
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
The polygon so described has 10 central angles with correspondingly 10 isosceles triangles. Side length of the polygon is base of the triangle, and the central angle is between each equal triangle side.
One-tenth of the total given area is the area of each isosceles triangle, 200%2F10=20 cm%5E2.

Central Angle value: 360%2F10+=+36 degrees.
Each of the equal angles of isosceles: %28180-36%29%2F2=72 degrees.

PROCEDURE TO PRODUCE EQUATIONS:
Draw altitute from the tip of triangle to middle of base. Label the altitude, h and half the base, %281%2F2%29b. Opposite of %281%2F2%29b is angle of 18 degrees. h is opposite of a 72 degree angle.


LAW OF SINES gives sin%2818%29%2F%28%281%2F2%29b%29=sin%2872%29%2Fh;

and area formula gives %281%2F2%29b%2Ah=20


SOLVING:
Those two equations are enough to solve for the altitude and the base of the triangle, this base being the side length of the 10-sided polygon. Having b and h, use pythorean theorem to find the other triangle side which will be the distance from center of the polygon to a vertex.
Note that each isosceles triangle here is composed of two right-trianlges each having angles 18, 72, 90.