Perimeter and Area

1. Perimeter and Area by Scott Alan Blanchard June 11, 2012

2. Where Do YOU need to go? Perimeter Area, defined Area of Squares and Rectangles Area of TRIANGLES All Done!! Feeling Good!

3. Perimeter Perimeter is a word used to describe “the distance around” a figure. Here’s a website with a little bit more information and help for perimeter http://www.mathleague.com/help/geometry/area.htm#perimeter Here’s a website with a little bit more information and help for perimeter http://www.mathleague.com/help/geometry/area.htm#perimeter

4. Perimeter When we are talking about perimeter we are talking about shapes that have non-curved lines. So we are talking about triangles, rectangles, squares, etc.

5. Here’s a square and let’s just say that each side has a length of 5 cm. A B D C

6. In other words, line segment AB is 5cm, line segment BC is 5 cm, line segment CD is 5 cm, and line segment DA is 5 cm. A B D C

7. To find its perimeter, we would just add up the measurement of each of its side. So 5 + 5 + 5 + 5 = 20 cm. A B D C

8. And here’s a rectangle… say it’s a football field, with a length of 100 yards and width of 50. 100 yards 50 yards 50 yards 100 yards

9. To find its perimeter, think of walking “all the way around” the field… 100 yards 50 yards 50 yards 100 yards

10. …you could start at one corner and walk 100 yards down and then 50 yards up, then 100 yards back and 50 yards back to where you started. 100 yards 50 yards 50 yards 100 yards

11. That would be a perimeter of 300 yards!! 100 yards 50 yards 50 yards 100 yards

12. Think you’ve got the hang of perimeter? It’s really not too hard. It’s the same for all non-curved geometric shapes. Let’s try a “TRY-ANGLE” now, O.K.?

13. Let’s give this triangle the following sides AB 15 cm, BC 22 cm, and AC 17 cm. See B A C

14. In your head, figure out what you think the perimeter would be for this triangle. Keep that number in your head and then hit next. AB 15 cm, BC 22 cm, and AC 17 cm. B A C

15. To find the perimeter of a triangle, simply add the sides together…. 15 22 + 17 = 54 cm. Is that what you got? I hope so! B A C

16. Another way to think of perimeter, is to think of it as if you were an ant and you had to walk “all the way around” the object. How far would you have to walk?

17. Just like when you walk somewhere, you ask how far away it is, perimeter, is pretty similar to that.

18. Can you think of a rectangle, square or triangle you are looking at right now?? There is probably more than one to see in the room you are in right now.

19. The computer screen is probably in the shape of a rectangle. Right?

20. Let’s find “its” perimeter. Are you ready?

21. Let’s say this is your computer!!Pretty cool, huh?!? Let’s give it a 15” wide and 10” high monitor. So what would its perimeter be?

22. We gave “YOUR” laptop a 15” wide and 10” high monitor. So what would its perimeter be? Remember *50 centimeters now that *45 inches perimeter *50 inches equals *60 inches the distance around…

23. ??ps. You are not using the right measurement. TRY AGAIN!!

24. 15 +10 +15 +10 ? 45 TRY AGAIN

25. What?!?!?!? My dog can count better than that!! 15 10 15 +10 Now YOU are trying to tell me that this is 60???? Ha, ha, ha, ha,…that’s good!! TRY AGAIN

26.  GREAT!! Now try this one.. Here’s a square CDEF. Each side has a length of 7 mm. What is its perimeter? C D E F

27. Auggh!! Be sure to check your adding and your units of measurement!!

28.  GREAT!! Now, let’s try this isosceles triangle. Isosceles triangles have two sides that have the same measure, and one that is longer or shorter than the others. Q R S 

29.  In this triangle, QRS, QR and QS will have lengths of 10 inches and RS will have a length of 15inches. What is the perimeter of triangle QRS?  Q R S 

30. Come on, NOW, don’t guess!! THINK!! Check your addition and units!

31. GREAT Job!! You remembered to add up the measurements of all the sides to get the perimeter!! Congratulations!! YOU are a “Powerful Perimeter-er!!”

32. Congratulations!! YOU are now a Perfect Perimeter Person!!

33. AREA Here’s a good website to go to to get more information and help with area. http://www.mathleague.com/help/geometry/area.htm#area Here’s a good website to go to to get more information and help with area. http://www.mathleague.com/help/geometry/area.htm#area

34. The “official” definition of area, as it pertains to mathematics, from Merriam-Webster is “the surface included within a set of lines; specifically : the number of unit squares equal in measure to the surface”. Pretty boring, huh? On its own it is, but we’ll get into what all that means in this exploration. Are you ready? Let’s go………………………………………………..

35. Area is more easily defined as “a measure of the space inside a closed, two-dimensional figure.” Here are some good websites for help on area: http://www.mathleague.com/help/geometry/area.htm#area http://www.brainpop.com/math/measurement/areaofparallelograms/index.weml?&tried_cookie=true http://www.brainpop.com/math/measurement/areaofpolygons/index.weml Here are some good websites for help on area: http://www.mathleague.com/help/geometry/area.htm#area http://www.brainpop.com/math/measurement/areaofparallelograms/index.weml?&tried_cookie=true http://www.brainpop.com/math/measurement/areaofpolygons/index.weml

36. By being “closed” it means that the shape comes to end where it started. You stop drawing a circle where you start it. Here are some good websites for help on area: http://www.mathleague.com/help/geometry/area.htm#area http://www.brainpop.com/math/measurement/areaofparallelograms/index.weml?&tried_cookie=true http://www.brainpop.com/math/measurement/areaofpolygons/index.weml Here are some good websites for help on area: http://www.mathleague.com/help/geometry/area.htm#area http://www.brainpop.com/math/measurement/areaofparallelograms/index.weml?&tried_cookie=true http://www.brainpop.com/math/measurement/areaofpolygons/index.weml

37. By being two-dimensional, the figures we will be dealing with are basically “flat”…drawn on paper. Here are some good websites for help on area: http://www.mathleague.com/help/geometry/area.htm#area http://www.brainpop.com/math/measurement/areaofparallelograms/index.weml?&tried_cookie=true http://www.brainpop.com/math/measurement/areaofpolygons/index.weml Here are some good websites for help on area: http://www.mathleague.com/help/geometry/area.htm#area http://www.brainpop.com/math/measurement/areaofparallelograms/index.weml?&tried_cookie=true http://www.brainpop.com/math/measurement/areaofpolygons/index.weml

38. In essence, the area of a shape is how many “square units” fit inside of that shape.

39. In the shapes below, if you counted how many squares were in each, you would know its area in square units.

40. You might be saying, “Hey, wait a minute!! The triangle and circle have ‘parts’ of squares in them!” Well, actually, you are partly correct.

41. In mathematics, everything ultimately works out “just right.” The ‘little bits’ of leftover squares combine and make squares. And we get a “final answer,” as the game show says.

42. With this rectangle we can squint real hard and count 20 full squares from side to side and 24 full squares from top to bottom. But on the very top and on the left side, it is obvious that we have anything but “full squares.”

43. What I was talking about earlier is that those will get combined and get taken care of.

44. When finding perimeter, we just needed to count the “distance around.” For area, we count the “amount inside.” So we are going to “count” the number of squares (or square units) inside.

45. The square units has to do with whether we are dealing with inches, millimeters, centimeters, kilometers, etc. etc… Whatever measure of units we are measuring our flat 2 dimensional object in.

46. I said we were going to count the square units inside the object. Well, not really, because we are going to learn a rule for area …multiply the length times the base….the number from side to side times the number from top to bottom.

47. The rule for finding area of a square or rectangle is A = H X W H W W

48. I said earlier that in this rectangle there are 20 squares from side to side and 24 from top to bottom. So now, according to that rule, we multiply 20 and 24 and get 480. Right? 24=H W=20 W

49. Wrong!!

50. Wrong!? You say, “How come?!?” 20 X 24 = 480. Well, yes, it does equal 480,sort of. You forgot your units!

51. You say, “You didn’t give us any units!” Well, then your answer is stated in just units, then. But when we deal with area, our units are squared, since we are counting the number of little squares.

52. So the correct answer is 480 square units or 480 units squared, or you may even see it written this way: 480 units2.

53. And if you wanted to take the time to, you could count and find there are 480 small squares in there. (you might have to put parts of squares together to make whole squares)

54. The basic formula for area for a square or rectangle is Area = Base X Height For example let’s say that the base of this rectangle is 50 mm, and its height is 18 mm. Then its area would be 50 mm X 18 mm = 900 mm2.

55. Here, let’s say that the base of this rectangle is 70 mm, and its height is 20 mm. Then its area would be 70 mm X 20 mm = 1400 mm2.

56. Why don’t you try?

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58. Why don’t you try a “word problem”? What would the area be for a rectangle with a base of 7 mm and height of 2 mm? 9 mm 14cm2 14mm2

59. Think! Think ! Think!

60. SUPER!! Here’s another one: A rectangle with a base of 12 cm and height of 5 cm would have an area of? 50 cm2 60 cm2 60 cm

61. Auggh!!! Be careful. Be VERY, VERY careful! Try again, please!

62. Waaaay to GOOO!! Give a try at this one now! What would the area of this rectangle be?

63. Please keep trying. I just know YOU can do it!! Area = Base X Height (in units squared)

64. Super Job!! Remember the formula for the area of a rectangle is Base X Height

65. We would find the area for a square the same way, because really a square is a rectangle. (But YOU already knew that!!) Right????

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71. Okay!! What’s the area? 5cm 5cm 10cm

72. Come on, now, I know you can do it!!

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75. Area of Triangles

76. Now, we’re going to learn about how to find the area of a triangle. It is a little different than finding the area of a square or rectangle.

77. For perimeter it was pretty easy…just add up all of the lengths of the sides. It didn’t matter. Rectangle, square or triangle. But what now, for area??

78. Believe it or not, a triangle is ½ of a square or rectangle. So it’s really not that hard, as long as you remember how to find the area of a square or rectangle.

79. Look at this rectangle below… What would happen if we cut it diagonally?

80. We’d end up with TWO triangles!! The same would happen with a square.

81. Here are some squares with diagonals to make triangles:

82. Since we have just seen that a triangle is ½ the size of a rectangle or square, what do you think is the way to find the area of a triangle??

83. Did you come up with something like this? ½ (Area of Rectangle) ?or? ½ (Area of Square) If so, GOOD for YOU!!

84. So, the formula would be, Area of a Triangle = ½ X BXH

85. It looks like this… ½(BXH)

86. If the height were ten, and the length were twelve of this triangle then its area would be ½ of 10 X 12….

87. …or ½ of 120. ½ of 120 is…. 60. So in this example, the area would be 60. 60 units squared.

88. Let’s try some out now. Okay?!? Remember the Area of a triangle is found by taking ½ of the product of the base and height! 12 inches 12 inches

89. Auggh!! Don’t forget the formula for the Area of a Triangle is ½ (B X H)!!

90. Oh, soooo close!! Try again!

91. W?W!! Great Job!! That’s right!! ½ of 12 inches X 12 inches is 72 inches2! Keep it up.

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95. Here’s another one. Try-- “your angle” --angle at this one! What’s the area?

96. Yahoo!! Ride ’em cowboy!!

97. Please try again. Your answer is INCORRECT!!

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99. Congratulations!! YOU feel confident on the topics of perimeter and area and this review is not necessary for you at this time. Good job for you. Come see your teacher for further instructions on what to do now…