1. Area: Formulas and Derivations
This slide show was written by
Michael Braverman
Trenton, NJ
March 2015

2. Area: Formulas and Derivations
A = bh h b
All area formulas are based on the formula of a rectangle:
Area of a rectangle = base x height.

3. Area: Why do we use square units?

4. Area: Why do we use square units?

5. Area: Why do we use square units?
This has an area of 12 squares, one unit on each side. These are called unit squares and have an area of 1 square unit (1 u2) each.

6. Area: How do we show work?
1. Begin with a General Formula
A General Formula is a rule that uses variables to represent parts that will always be there.
3 units
4 units

7. Area: How do we show work?
1. Begin with a General Formula
The General Formula (when applied correctly) ALWAYS gives the correct answer.
3 units
4 units

8. Area: How do we show work?
1. Begin with a General Formula
Area of a Rectangle = base x height
3 units
4 units

9. Area: How do we show work?
1. Begin with a General Formula
Area of a Rectangle = base x height
Note: base and height are better terms to use than length and width because of the way the other formulas work.
3 units
4 units

10. Area: How do we show work?
1. Begin with a General Formula
Area of a Rectangle = base * height
Abbreviate this as:
A = b * h
3 units
4 units

11. Area: How do we show work?
2. Substitute in the information that you know.
A = b * h
A = 4u * 3u
NOTE: Keep your units throughout your work to prevent careless errors! 3u 4u

12. Area: How do we show work?
3. Simplify. Be sure to write your units correctly!
A = b * h
A = 4u * 3u
A = 12u2 3u 4u

13. Area of a Parallelogram
A = bh
5 units
4 units
8 units

14. Area of a Parallelogram
A = bh
5 units
4 units
8 units

15. Area of a Parallelogram
A = bh
5 units
4 units
8 units

16. Area of a Parallelogram
A = bh
5 units
4 units
8 units

17. Area of a Parallelogram
A = bh
5 units
4 units
8 units

18. Area of a Parallelogram
A = bh
5 units
4 units
8 units

19. Area of a Parallelogram
A = bh
5 units
8 units
4 units

20. Area of a Parallelogram
A = bh
5 units
8 units
4 units

21. Area of a Parallelogram
A = bh
5 units
8 units
4 units

22. Area of a Parallelogram
A = bh
5 units
4 units
8 units

23. Area of a Parallelogram
4 units
5 units
8 units

24. Area of a Parallelogram
4 units
5 units
8 units

25. Area of a Parallelogram
A = bh
4 units
5 units
8 units

26. Area of a Parallelogram
A = bh
5 units
4 units
8 units

27. Area of a Parallelogram
A = bh
5 units
4 units
8 units

28. Area of a Parallelogram
A = bh
5 units
4 units
8 units

29. Area of a Parallelogram
A = bh
A = 8u * 4u
5 units
4 units
8 units

30. Area of a Parallelogram
A = bh
A = 8u * 4u
5 units
4 units
A = 32u2
8 units
Note that the “5 units” is NOT used in the area at all. (It is part of the original perimeter)

31. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

32. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

33. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

34. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

35. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

36. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

37. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

38. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

39. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

40. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

41. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

42. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

43. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

44. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

45. Area of a Triangle
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

46. Area of a Triangle This is a copy of the bottom side.
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

47. Area of a Triangle This is a copy of the left side.
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

48. Area of a Triangle This “side” is Congruent to itself.
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

49. Area of a Triangle Similarly: These Angles are congruent.
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

50. Area of a Triangle Similarly: These So are these.
Angles are congruent.
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

51. Area of a Triangle Similarly: These And these. Angles are congruent.
All triangles can be “cloned” and “flipped” to form a parallelogram. This means that the area of a triangle is 1/2 the area of a parallelogram.

52. Area of a Triangle
Therefore, the area of a triangle = the area of a parallelogram divided by 2.

53. Area of a Triangle
Therefore, the area of a triangle = the area of a parallelogram divided by 2.

54. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

55. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

56. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

57. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

58. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

59. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

60. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

61. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

62. Area of a Trapezoid Base 1 height Base 2
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

63. Area of a Trapezoid Base 1 Base 2 height Base 2 Base 1
The area of a trapezoid can be found the same way, but the formula becomes more complicated.

64. Area of a Trapezoid Base 1 Base 2 height Base 2 Base 1

65. This is the AVERAGE of the two bases!
Area of a Trapezoid
Base 1
Base 2
height
Base 2
Base 1
Area of a trapezoid =
This is the AVERAGE of the two bases!

66. Area of a Circle

67. Area of a Circle
Circumference
diameter
radius

68. Area of a Circle In a circle, the Diameter is always equal
Circumference
In a circle, the
Diameter is always equal
to two times the radius
d = 2r
In a circle, the
circumference
diameter
is always a little
greater than 3.
diameter
radius
This ratio of the
circumference to the diameter of a circle
Is called  and
is approximately equal to 3.14

69. Area of a Circle

70. Area of a Circle

71. Area of a Circle

72. Area of a Circle

73. Area of a Circle

74. Area of a Circle

75. Area of a Circle

76. Area of a Circle

77. Area of a Circle

78. Area of a Circle

79. Area of a Circle

80. Area of a Circle

81. Area of a Circle

82. Area of a Circle

83. Area of a Circle

84. Area of a Circle

85. Area of a Circle

86. Area of a Circle

87. Area of a Circle

88. Area of a Circle

89. Area of a Circle

90. Area of a Circle

91. Area of a Circle

92. Area of a Circle

93. Area of a Circle

94. Area of a Circle
radius

95. Area of a Circle
radius
This shows a circle broken up into ONLY 8 pieces. Suppose, we broke it up into a lot more!

96. Area of a Circle
radius

97. Area of a Circle
radius

98. Area of a Circle
radius

99. Area of a Circle
radius