This pdf contains the area of a regular polygon which is one-half the product of its apothem and its perimeter. Often the formula is written like this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the perimeter
1. Areas of Regular Polygons
2. Lesson Focus The focus of this lesson is on applying the formula for finding the area of a regular
3. 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
4. Basic Terms
5. Theorem 11-11 The area of a regular polygon is equal to half the product of the apothem and the
6. Area of a regular polygon The area of a regular polygon is: A = ½ Pa Area Perimeter apothem
7. The center of circle A is: A B The center of pentagon BCDEF is: A F C A radius of circle A is: A AF A radius of pentagon BCDEF is: G AF E D An apothem of pentagon BCDEF is: AG
8. 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 The number of congruent triangles formed will be A = ½ aP, or A = ½ a • ns. 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
9. 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
10. 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. 3 2 360/5 = 72 ½ (72) = 36 L2 = 36 1 • L1 = 72 L3 = 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 in2 Example 3: Find the area. 10 mm A=½ap p = 60 mm • LL = √3 • 5 = 8.66 a 5 mm A = ½ • 8.66 • 60 A = 259.8 mm2
11. • But what if we are not given any angles.
12. Ex: A regular octagon has a radius of 4 in. Find its area. First, we have to find the 67.5 o x apothem length. a x 4 sin 67.5 cos 67.5 4 4 a 3.7 4cos67.5 = x 4sin67.5 = a 135o 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 in2
13. 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 360 central is: n Y Ex: Find mY. 72o
+1-315-636-4485
+91 98151-74050 
