The document discusses formulas for finding the area of regular polygons based on their apothem and perimeter. The apothem is the distance from the polygon's center to its midpoint side. For squares, the area formula is A = ps/2. For equilateral triangles, the formula is A = (sqrt(3)s^2)/4. For regular hexagons, the formula is A = (3sqrt(3)s^2)/2. It provides examples of using these formulas to find apothems, perimeters, and areas.

1. Area of Regular Polygons

2. The Apothem The apothem (a) is the segment drawn from the center of the polygon to the midpoint of the side (and perpendicular to the side)

3. Deriving the Formula - Squares The diagonals of the square divide it into four triangles with base s and height a. The area of each triangle is sa. Since there are 4 triangles, the total area is 4( )sa or (4s)a. Since the perimeter (p) = 4s the formula becomes A = ap

4. Deriving the Formula - Triangles Connecting the center of the equilateral triangle to each vertex creates three congruent triangles with area A = sa. Since there are three triangles, the total area is 3( )sa, or (3s)a. Since the perimeter = 3s, the formula may be written A = ap

5. Deriving the Formula - Regular Hexagons Connecting the center of the regular hexagon to each vertex creates six congruent triangles with area A = sa. Since there are six triangles, the total area is 6( )sa, or (6s)a. Since the perimeter = 6s, the formula may be written A = ap

6. Finding the apothem - Square The apothem of a square is one-half the length of the side. If s = 15, a = ? a = 7.5 If a = 14, s = ? s = 28

7. Find the apothem - Triangles The apothem of an equilateral triangle is the short leg of a 30-60-90 triangle where s/2 is the long leg. 30  60  90  Then a = (s/2)/  3 or

8. Find the apothem - Triangles If s = 18, a = ? If s = 24, a = ? If s = 10, a = ?

9. Find the side - Triangles If the apothem is 6 cm, the side = ? If the apothem is 2.5 cm, the side = ?

10. Finding the apothem - Hexagons The apothem of a regular hexagon is the long leg of a 30-60-90 triangle. 60  90  30  Therefore, the apothem is (s/2)

11. Finding the apothem - hexagons If the side = 12 the apothem = ? If the side = 5 the apothem = ?

12. Finding the side - hexagons If the apothem = 12, the side = ? If the apothem = 16, the side = ?

13. Finding the area - Squares a = 6 cm Find the area A = 144 cm 2 A = 288 cm 2 A = 50 cm 2

14. Finding the Area - Triangles If a = 3 cm, find the area of the triangle

15. Finding the Area - Triangles If a = 5 cm, find the area of the triangle

16. Finding the Area - Triangles

17. Finding the Area - Triangles If the side of the triangle = 10 cm, find the area of the triangle

18. Finding the Area - Triangles

19. Finding the area - hexagons If the a = 6 cm, find the area of the hexagon.

20. Finding the area - hexagons

21. Finding the area - hexagons

22. Finding the area - hexagons