Area of Regular Polygons

Terms Radius – segment joining the center of the polygon to the vertex of the polygon. All radii of a polygon are equal. When all radii are drawn, congruent triangles are formed and the sum of the angles around the center is equal to 360° m  1 + m  2 + m  3 + m  4 +m  5 = 360° So, m  4 = 360 ÷ 5 = 72°

Terms 72° Apothem – the apothem is the segment from the center Perpendicular to a side of the regular polygon. The apothem creates a right triangle with HALF the side and radius of the polygon and it cuts the central Angle in half. You can use trig to find the apothem if you know the Length of the sides of the polygon. 36° a ½ s Tan 36° = ½ S a a = ½ s ÷ tan 36

Apothem Formula Area = ½ a P or Area = ½ a n s 72° ° a Tan 36° = 9 / a a = 9 ÷ tan 36° Area = ½ a n s = ½ (9 / tan 36) (5)(18)  square units

Apothem Formula Area = ½ a P or Area = ½ a n s 72° s ½ s 36° 9 Tan 36° = ½ s / 9 ½ s = 9 (tan 36°) s = 18(tan 36°) Area = ½ a n s = ½ (9)(5)(18 tan 36°)  square units What if you are given the apothem but not the side length?

Apothem Formula Area = ½ a P or Area = ½ a n s 72° s ½ s 36° 5 Sin 36° = ½ s / 5 ½ s = 5 (sin 36°) s = 10 (sin 36°) Area = ½ a n s = ½ (5cos36°)(5)(10 sin 36°)  59.4 square units What if you are given the radius….not the apothem or side? a cos 36° = a / 5 a = 5 (cos 36°) s = 10 (sin 36°) a = 5 (cos 36°)

Worksheet 11.2 A 45° 4 ½ s   Given: 4 sides, apothem = 4 ½ s = 4 s = 8 n = 4 a = 4 Area = ½ a n s = ½(4)(4)(8) = 64 units 2

Worksheet 11.2 A 30° 4 ½ s   Given: Regular hexagon, radius = 4 ½ s = 2 s = 4 n = 6 a = 2  3 Area = ½ a n s = ½(2  3)(6)(4) = 24  3 units 2  41.6 units 2 360° ÷ 6 = 60° 60° ÷ 2 = 30° a

Worksheet 11.2 A 36° ½ s 14. Given: Regular pentagon, apothem = 4 a = 4 n = 5 ½ s = 4 tan 36° s = 8 tan 36° Area = ½ a n s = ½(4)(5)(8 tan 36°) = 80 tan 36° units  58.1 units 2 360° ÷ 5 = 72° 72° ÷ 2 = 36° 4 Tan 36° = ½ s 4

Worksheet 11.2 A 30° ½ s 15. Given: Regular hexagon, apothem = 2  3 a = 2  3 n = 6 ½ s = 2 s = 4 Perimeter = ns = 6(4) = 24 Area = ½ a n s = ½(2  3)(6)(4) = 24  3 units 2  41.6 units 2 360° ÷ 6 = 60° 60° ÷ 2 = 30° 232   3 4

Worksheet 11.2 A 22.5° ½ s 17. Given: Regular octagon, apothem = 8 a = 8 n = 8 s = 16 tan22.5° Perimeter = ns = 8(16tan22.5°)  53.0 units Area = ½ a n s = ½(8)(8)(16tan22.5°)  units 2 8 Tan 22.5° = ½ s 8