Areas of Regular Polygons Lesson Focus The focus

Areas of Regular Polygons

Lesson Focus The focus of this lesson is on applying the formula for finding the area of a regular polygon.

Basic Terms Center of a Regular Polygon the center of the circumscribed circle Radius of a Regular Polygon the distance from the center to a vertex Central Angle of a Regular Polygon an angle formed by two radii drawn to consecutive vertices Apothem of a Regular Polygon the (perpendicular) distance from the center of a regular polygon to a side

Basic Terms

Theorem 11 -11 The area of a regular polygon is equal to half the product of the apothem and the perimeter.

Area of a regular polygon The area of a regular polygon is: A = ½ Pa Area Perimeter apothem

B F A G E D The center of circle A is: A The center of pentagon BCDEF is: A C A radius of circle A is: AF A radius of pentagon BCDEF is: AF An apothem of pentagon BCDEF is: AG

Area of a Regular Polygon • The area of a regular n-gon with side lengths (s) is half the product of the apothem (a) and the perimeter (P), so A = ½ a. P, or A = ½ a • ns. The number of congruent triangles formed will be the same as the number of sides of the polygon. NOTE: In a regular polygon, the length of each side is the same. If this length is (s), and there are (n) sides, then the perimeter P of the polygon is n • s, or P = ns

More. . . • A central angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. You can divide 360° by the number of sides to find the measure of each central angle of the polygon. • 360/n = central angle

Areas of Regular Polygons Center of a regular polygon: center of the circumscribed circle. Radius: distance from the center to a vertex. Apothem: Perpendicular distance from the center to a side. Example 1: Find the measure of each numbered angle. L 2 = 36 ½ (72) = 36 360/5 = 72 L 1 = 72 3 2 1 • L 3 = 54 Area of a regular polygon: A = ½ a p where a is the apothem and p is the perimeter. Example 2: Find the area of a regular decagon with a 12. 3 in apothem and 8 in sides. Perimeter: 80 in A = ½ • 12. 3 • 80 A = 492 in 2 Example 3: Find the area. A=½ap 10 mm p = 60 mm LL = √ 3 • 5 = 8. 66 A = ½ • 8. 66 • 60 • a A = 259. 8 mm 2 5 mm

• But what if we are not given any angles.

Ex: A regular octagon has a radius of 4 in. Find its area. 67. 5 o x First, we have to find the apothem length. 4 a 3. 7 135 o 4 cos 67. 5 = x 4 sin 67. 5 = a 3. 7 = a 1. 53 = x Now, the side length. Side length=2(1. 53)=3. 06 A = ½ Pa = ½ (24. 48)(3. 7) = 45. 288 in 2

Last Definition Central of a polygon – an whose vertex is the center & whose sides contain 2 consecutive vertices of the polygon. Y is a central . Measure of a central is: Y Ex: Find m Y. 360/5= 72 o

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