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.
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.