Areas of Polygons

1. Areas of Polygons Parallelograms, Triangles, and Circles

2. CCS: • 6.G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems

3. Objectives Students will be able to: • Find the areas of parallelograms, triangles, and circles • Find the circumference of circles • Find the area of complex figures

4. AREAS OF POLYGONS QUICK REVIEW: What is the formula to find the area of a rectangle? A = l x w

5. AREA OF A PARALLELOGRAM b h Let’s Discover the formula for a parallelogram!

6. AREA OF A PARALLELOGRAM b h To do this let’s cut the left triangle and…

7. AREA OF A PARALLELOGRAM b h h slide it…

8. AREA OF A PARALLELOGRAM h b h Keep Sliding…..

9. AREA OF A PARALLELOGRAM h b h Keep Sliding…..

10. AREA OF A PARALLELOGRAM h b h Keep Sliding…..

11. b h AREA OF PARALLELOGRAM …thus, changing it to a rectangle. What is the area of the rectangle?

12. AREA OF A PARALLELOGRAM b h Since the area of the rectangle and parallelogram are the same, just rearranged, what is the formula for the area of this parallelogram?

13. Area of a Parallelogram • Any side of a parallelogram can be considered a base. The height of a parallelogram is the perpendicular distance between opposite bases. • The area formula is A=bh A=bh A=5(3) A=15m2

14. Video Time • Area of a parallelogram

15. AREA OF A TRIANGLE Now we will discover the formula for area of a triangle. h b

16. ? ? b AREA OF A TRIANGLE Let’s divide the triangle so that we divide the height in two.

17. ? ? b AREA OF A TRIANGLE Remember, we divided the height into two equal parts. • Now take the top and rotate…

18. AREA OF A TRIANGLE ? rotate… ?

19. AREA OF A TRIANGLE ? rotate… ? b

20. AREA OF A TRIANGLE ? rotate… ? b

21. AREA OF A TRIANGLE ? ? rotate… b

22. ? ? b AREA OF A TRIANGLE …until you have a parallelogram. How would you represent the height of this parallelogram?

23. AREA OF A TRIANGLE ? ? ? ? b b Remember, you divided the height in two.

24. AREA OF A TRIANGLE ? b What is the area of this parallelogram?

25. AREA OF A TRIANGLE The area of this triangle would be the same as the parallelogram. Therefore, the formula for the area of a triangle is… what? h b

26. Example A= ½ bh A= ½ (30)(10) A= ½ (300) A= 150 km2

27. Video Time • Finding the area of triangles

28. Now Let’s Talk Circles…. The circumference of a circle is The distance around a circle Hint: Circumference remember circle around

29. What is the formula relating the circumference to the diameter? circumference centre Diameter Radius

30. C = ? x d People knew that the circumference is about 3 times the diameter but they wanted to find out exactly. C ≈ 3 x d This means APPROXIMATELY EQUAL TO

31. Early Attempts Egyptian Scribe Ahmes. in 1650 B.C. said C≈3.16049 x d Archimedes, said C ≈3.1419 x d Fibonacci. In 1220 A.D. said C≈3.1418xd What is the value of the number that multiplies the diameter to give the circumference????

32. The exact true value is…………… UNKNOWN!!

33. An approximation to π π≈3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609................forever….

34. Circumference • Remember, circumference is the distance around the circle. • If you divide a circle’s circumference by its diameter, you always get the same irrational number – pi (symbol: ) • This is true of every circle. • We estimate pi to be 3.14 or the fraction 22/7.

35. Circumference Formulas C = d C = 2r

36. Example C = d C = (3.14)(41) 41 m C = 128.74 m We substitute 3.14 in for pi.

37. Finding the Area of a Circle • The formula for the area of a circle is • We say: Area = pi times radius squared

38. Example A = r2 A = (3.14)(82) 8 mm A =(3.14)(64) A = 200.96 mm2

39. Example 2 If you are given a diameter, divide it in half to find the radius. 13 divided by 2 equals 6.5 cm. A = r2 13 cm A = (3.14)(6.52) A = (3.14)(42.25) A = 132.665 cm2

40. A circle is defined by its diameter or radius The Area and Perimeter of a Circle The perimeter or circumference of a circle is the distance around the outside radius The area of a circle is the space inside it Diameter The ratio of π (pi) The circumference is found using the formula C=π d or C= 2πr (since d=2r) The area is found using the formula A=πr2

41. Video Time • Finding circumference • Naming the parts of a circle • Finding the area of circles

42. Complex Figures • Use the appropriate formula to find the area of each piece. • Add the areas together for the total area.

43. Example 24 cm 10 cm | 27 cm | Split the shape into a rectangle and triangle. The rectangle is 24cm long and 10 cm wide. The triangle has a base of 3 cm and a height of 10 cm.

44. Solution Rectangle Triangle A = ½ bh A = lw A = ½ (3)(10) A = 24(10) A = 240 cm2 A = ½ (30) A = 15 cm2 Total Figure A = A1 + A2 A = 240 + 15 = 255 cm2

45. Classwork: • Try This Area of Parallelograms Game- You have to be QUICK!! • Try This Baseball Game that finds area of triangles Homework: Reteaching/Practice 9.4 HO