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

4. You can divide a regular polygon into congruent isosceles triangles by drawing segments from the center of the polygon to each vertex. 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. Areas of Regular Polygons

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

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? ½ as What is the area of this pentagon in terms of a and s? Areas of Regular Polygons

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? 6.½ as What is the area of this heptagon in terms of a and s? 7.½ as Areas of Regular Polygons

8. 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 segment as the apothem. Areas of Regular Polygons The apothem is the height of a triangle between the center and two consecutive vertices of the polygon. Apothem a

9. 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 segment as the apothem. Areas of Regular Polygons The apothem is the height of a triangle between the center and two consecutive vertices of the polygon. Apothem a G F E DC B A H Hexagon ABCDEF with center G, radius GA, and apothem GH

10. n = number of sides s = base Areas of Regular Polygons 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: 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 where …. A is the area, P is the perimeter, a is the apothem, s is the length of each side, and n is the number of sides. Areas of Regular Polygons or

12. Find the unknown length accurate to the nearest unit, or the unknown area accurate to the nearest square unit. Recall that the symbol  is used for measurements or calculations that are approximations. A  ? s = 24 cm a  24.9 cm A  2092 cm 2 A  19,887.5 cm 2 s = 107.5 cm a  ? a  74 cm P  ? A = 4940.8 cm 2 a = 38.6 cm P  256 cm Areas of Regular Polygons Examples:

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

14. Another look... 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 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 This approach can be used to find the area of any regular polygon.