11 6 Areas of Regular Polygons Hubarth Geometry

11. 6 Areas of Regular Polygons Hubarth Geometry

Apothem of a polygon is the height of an isosceles triangle that has two radii as legs P N M Q Theorem 11. 11 Area of a Regular Polygon The area of a regular n-gon with side length s is one half the product of the apothem a and the perimeter p a s

Ex 1 Find Angle Measures in a Regular Polygons In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

Ex 2 Find the Area of a Regular Polygon You are decorating the top of a table by covering it with small ceramic tiles. The table top is a regular octagon with 15 inch sides and a radius of about 19. 6 inches. What is the area you are covering? Find the perimeter P of the table top. An octagon has 8 sides, so P = 8(15) = 120 inches. To find RS, use the Pythagorean Theorem for ∆ RQS.

Ex 2 continued Find the area A of the table top. 1 ≈ (18. 108)(120) 2 ≈ 1086. 5 So, the area you are covering with tiles is about 1086. 5 square inches.

Ex 3 Find the Perimeter and Area of a Regular Polygon A regular nonagon is inscribed in a circle with radius 4 units. Find the perimeter and area of the nonagon. sin 20° = MK LK cos 20° = LM LK sin 20° = MK 4 cos 20° = LM 4 4 sin 20° = MK 4 cos 20° = LM The regular nonagon has side length s = 2 MK = 2(4 sin 20°) = 8 sin 20° and apothem a = LM = 4 cos 20°.

Practice In the diagram, WXYZ is a square inscribed in P. 1. Identify the center, a radius, an apothem, and a central angle of the polygon. Find the perimeter and the area of the regular polygon. 3. 4. P= 46. 6 units, A = 151. 5 units 2 5. P = 70 units, A = 377. 0 units 2