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. 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