Lesson 8-4 Areas of Regular Polygons

1. Lesson 8-4 Areas of Regular Polygons

2. In this lesson you will… • ● Discover the area formula for regular polygons Areas of Regular Polygons

3. Let’s recall some concepts Polygon Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Areas of Regular Polygons Regular and Irregular Polygons If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

4. You can divide a regular polygon into congruent isosceles triangles by drawing segments from the center of the polygon to each vertex. Areas of Regular Polygons Center of a polygon The center of its circumscribed circle Radii of a polygon the radius of its circumscribed circle, or the distance from the center to a vertex.

5. If you divide regular polygons into triangles. Then you will be able to write a formula for the area of any regular polygon. To find the are of a triangle you use the following formula Areas of Regular Polygons

6. Consider a regular pentagon with side length s, divided into congruent isosceles triangles. Each triangle has a base s and a height a. What is the area of one isosceles triangle in terms of a and s? Areas of Regular Polygons ½ as What is the area of this pentagon in terms of a and s?

7. What is the area of this pentagon in terms of a and s? 5.½ as What is the area of this hexagon in terms of a and s? Areas of Regular Polygons 6.½ as What is the area of this heptagon in terms of a and s? 7.½ as

8. Apothem The apothem of a regular polygon is a perpendicular segment from the center of the polygon’s circumscribed circle to a side of the polygon. You may also refer to the length of the segmentas theapothem. Areas of Regular Polygons The apothem is the height of a triangle between the center and two consecutive vertices of the polygon. a

9. Apothem The apothem is the height of a triangle between the center and two consecutive vertices of the polygon. F A H Areas of Regular Polygons a The apothem of a regular polygon is a perpendicular segment from the center of the polygon’s circumscribed circle to a side of the polygon. You may also refer to the length of the segmentas theapothem. E G B D C Hexagon ABCDEF with center G, radius GA, and apothem GH

10. 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 in terms of n and s is: Areas of Regular Polygons n = number of sides s = base The number of congruent triangles formed will be the same as the number of sides of the polygon.

11. The area of a regular polygon is given by the following formulsa or Areas of Regular Polygons where…. Aisthearea, P is the perimeter, ais the apothem, s is the length of each side, and n is thenumber of sides.

12. Examples: Findtheunknownlengthaccuratetothenearestunit, ortheunknownareaaccuratetothenearestsquareunit. Areas of Regular Polygons A  19,887.5 cm2 s = 107.5 cm a  ? P  ? A = 4940.8 cm2 a = 38.6 cm A  ? s = 24 cm a  24.9 cm P  256 cm A  2092 cm2 a  74 cm Recallthatthe symbol isusedformeasurementsorcalculationsthat are approximations.

13. Find the approximate area of the shaded region of the regular polygon. • Find the area of the entire pentagon • Find the area of unshaded triangle 8.0 ft • Then subtract 5.5 ft

14. Another look... This approach can be used to find the area of any regular polygon. A = Area of 1 triangle • # of triangles A = ( ½ • apothem • side length s) • # of sides A = ½ • apothem • # of sides • side length s A = ½ • apothem • perimeter of a polygon