Chapter 14. Regular Polygons and the Circle. Regular Polygons. A regular polygon is a convex polygon that is both equilateral and equiangular. Regular hexagon. Regular pentagon. Equilateral triangle. square. Theorem : Every regular polygon is cyclic.
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Regular Polygons and the Circle
Draw the circle that contains points A, B, and C, and let its center be called O.
Draw OA, OB, OC, and OD.
OB=OC (all radii of a circle are equal)
Therefore, 1= 2 (if 2 sides of a triangle are equal, the are equal, the angles opposite them are equal)
Because ABC= BCD (a regular polygon is equiangular), 3= 4 (subtraction).
Also, AB = CD (a regular polygon is equilateral), and so ∆OBA =∆OCD (SAS)
Therefore, OD= OA
And because OA is the radius of the circle, it follows that OD also is a radius.
where p is the perimeter of the polygon
n is the number of sides
s is the length of one side
in which N= n(sin180/n)
and r is its radius
in which M= n(sin180/n)(cos180/n)
and r is its radius
because the diameter of a circle is twice its radius, it follows from the equations c=2r and d=2r that c=d
If the diameter of a circle is d, its circumference is d
If the radius of a circle is r, its area is r2
If the circle had a radius of 12 inches, and was cut into 8 equal pieces, the area of each piece is
1/8(12)2 =18 57 square inches
Because the border of the circle (360°) also is divided into eight congruent pieces, each sector has an arc (and central angle) with a measure of
The lengths of these eight equal arcs add up to give the circumference of the circle; so each arc has length
1/8(2)(12)= 3 9 inches
Because the arc of every circle has a measure of 360°, the area of the sector must be m/360 times the area of the circle, or m/360r2 .
The length of its arc is m/360 times the circumference of the circle, or m/360(2)r.
The idea is simple:
If a sector is a certain fraction of a circle, then its area is the same fraction of the circle’s area. If an arc is a certain fraction of a circle, then its length is the same fraction of the circle’s circumference.