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Regular Polygon Center and Central Angle of a Regular Polygon Radius and Apothem of a Regular Polygon More About Regular Polygons 7.3 KEY CONCEPTS Several interesting properties of regular polygons are developed in this section. For in- stance, every regular polygon has both an inscribed circle and a circumscribed circle; fur- thermore, these two circles are concentric. In Example 1, we use bisectors of the angles of a square to locate the center of the inscribed circle. The center, which is found by using the bisectors of any two consecutive angles, is equidistant from the sides of the square. Reminder A regular polygon is both equilateral and equiangular. EXAMPLE 1 Given square ABCD in Figure 7.24a, construct inscribed . } O A D B C a Figure 7.24 A D B C b O c A M D B C O Solution Figure 7.24b: The center of an inscribed circle must lie at the same distance from each side. Center O is the point of concurrency of the angle bisectors of the square. Thus, we construct the angle bisectors of and to identify point O. Figure 7.24c: Constructing , OM is the distance from O to and the length of the radius of the inscribed circle. Finally we construct inscribed with radius as shown. OM } O AB OM ⬜ AB ∠C ∠B 쮿 In Example 2, we use the perpendicular bisectors of two consecutive sides of a reg- ular hexagon to locate the center of the circumscribed circle. The center determines a point that is equidistant from the vertices of the hexagon. EXAMPLE 2 Given regular hexagon MNPQRS in Figure 7.25a, construct circumscribed . } X R Q P S M N a Figure 7.25 R Q P X S M N b R Q P X S M N c In Chapter 2, we saw that the sum of the measures of the interior angles of a polygon with n sides is given by . In turn, the measure I of each interior angle of a regular polygon of n sides is given by . The sum of the measures of the exterior angles of any polygon is always 360°. Thus, the measure E of each exterior I = n - 2180 n S = n - 2180 A B D C Figure 7.26 Solution Figure 7.25b: The center of a circumscribed circle must lie at the same distance from each vertex of the hexagon. Center X is the point of concurrency of the perpendicular bisectors of two consecutive sides of the hexagon. In Figure 7.25b, we construct the perpendicular bisectors of and to locate point X. Figure 7.25c: Where XM is the distance from X to vertex M, we use radius to construct circumscribed . } X XM NP MN 쮿 For a rectangle, which is not a regular polygon, we can only circumscribe the cir- cle see Figure 7.26. Why? For a rhombus also not a regular polygon, we can only inscribe the circle see Figure 7.27. Why? As we shall see, we can construct both inscribed and circumscribed circles for reg- ular polygons because they are both equilateral and equiangular. A few of the regular polygons are shown in Figure 7.28. H K L J Figure 7.27 Equilateral Triangle Figure 7.28 Square Regular Pentagon Regular Octagon Exs. 1–6 EXAMPLE 3 a Find the measure of each interior angle of a regular polygon with 15 sides. b Find the number of sides of a regular polygon if each interior angle measures 144°. Solution a Because all of the n angles have equal measures, the formula for the measure of each interior angle, becomes which simplifies to 156°. I = 15 - 2180 15 I = n - 2180 n angle of a regular polygon of n sides is . E = 360 n