SOLUTION: a regular pentagon is inscribed in a circle whose radius is 18cm. find the perimeter of the pentagon

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Question 888882: a regular pentagon is inscribed in a circle whose radius is 18cm. find the perimeter of the pentagon
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the radius of the circle is 18 cm.
this radius is also the equal sides of the isosceles triangle formed.
you want to find the length of the base of the triangle formed.
that will allow you to find the perimeter.
in order to find the length of the base, you need to find the interior angles of one of the triangles formed in the pentagon.
since all of the triangles formed are congruent in a regular pentagon, if you find one, you've found them all.

the central angle in each triangle formed from the center of the pentagon to the vertex of the pentagon is equal to 360 / n, where n = the number of sides, which is 5 in the case of a pentagon.

360 / 5 = 72 degrees.
that's the central angle of each of the triangles formed.

since the triangles are isosceles, the base angles are equal to (180 - 72) / 2 which makes each base angle of one of the triangles in the pentagon equal to 54.

drop a perpendicular from the central angle of one triangle to the base of that triangle and you form two congruent right triangles, each of which has a central angle of 36 degrees and a right angle of 90 degrees and a leg angle of 54 degrees.

the following picture shows you what i mean.
the pentagon is ABCDE.
one of the triangles formed is AOE.
the perpendicular that is dropped from the central angle is OF.
OAF and OEF are base angles of the two right triangles formed.
the central angle of 72 degrees is split into two angles of 36 degrees apiece.
The right triangle of OEF was used to calculate FE which is equal to one half the length of the side of the pentagon.
Two times that is the length of the side of the pentagon.
FE and AF are referenced as x in the equation.
AE is referenced as 2x in the equation.
x is equal to 10.58...
2x is equal to 21.16...
5 * 2x is equal to 105.80...
105.l80 is referenced as P.
that's your perimeter.