Biggie Size it

Investigation using

Microsoft Excel

 


 

Let’s create a Microsoft excel spreadsheet with our original problem’s side lengths:

 

 

Size

Side a

Side b

Side c

Perimeter

Area

1

1.7

2.3

3.9

 

 

2

3.4

4.6

7.8

 

 

3

5.1

6.9

11.7

 

 

4

6.8

9.2

15.6

 

 

5

8.5

11.5

19.5

 

 

6

10.2

13.8

23.4

 

 

7

11.9

16.1

27.3

 

 

8

13.6

18.4

31.2

 

 

9

15.3

20.7

35.1

 

 

10

17

23

39

 

 

 

 

The perimeter will be easy to compute, by just adding the sides together but the area will be a little harder to compute.  To compute the area given 3 side lengths instead of a base and height, we will have to explore Heron’s Formula.

 

Let’s first fill in the perimeter and explore our prediction that as the side lengths double, triple, etc, so will the perimeter.

 

Size

Side a

Side b

Side c

Perimeter

Ratio

1

1.7

2.3

3.9

7.9

 

2

3.4

4.6

7.8

15.8

2

3

5.1

6.9

11.7

23.7

3

4

6.8

9.2

15.6

31.6

4

5

8.5

11.5

19.5

39.5

5

6

10.2

13.8

23.4

47.4

6

7

11.9

16.1

27.3

55.3

7

8

13.6

18.4

31.2

63.2

8

9

15.3

20.7

35.1

71.1

9

10

17

23

39

79

10

 

This holds true.

 

Now let’s use Heron’s Formula to find the area of each triangle.

 

Heron’s Formula:

 

Heron's formula for the area of a triangle with sides of length a, b, c is

where

 

Here is the excel spreadsheet filled out:

 

 

 

 

 

 

Heron's Formula

 

 

Size

Side a

Side b

Side c

Perimeter

Ratio

s

s-a

s-b

s-c

Area

Ratio

1

1.7

2.3

3.9

7.9

 

3.95

2.25

1.65

0.05

0.86

 

2

3.4

4.6

7.8

15.8

2

7.9

4.5

3.3

0.1

3.43

4

3

5.1

6.9

11.7

23.7

3

11.85

6.75

4.95

0.15

7.71

9

4

6.8

9.2

15.6

31.6

4

15.8

9

6.6

0.2

13.70

16

5

8.5

11.5

19.5

39.5

5

19.75

11.25

8.25

0.25

21.41

25

6

10.2

13.8

23.4

47.4

6

23.7

13.5

9.9

0.3

30.83

36

7

11.9

16.1

27.3

55.3

7

27.65

15.75

11.55

0.35

41.96

49

8

13.6

18.4

31.2

63.2

8

31.6

18

13.2

0.4

54.80

64

9

15.3

20.7

35.1

71.1

9

35.55

20.25

14.85

0.45

69.36

81

10

17

23

39

79

10

39.5

22.5

16.5

0.5

85.63

100

 

If we graph the relationship of the size of each triangle being doubled, tripled, etc. we get the below graph:

 

Notice it is as we figured; a linear relationship.


Now we will do the same with the area.

 

Just as we suspected it is a squared function.

Let’s continue to increase the size of the triangles and graph our data.

 

 

 

 

 

 

Heron's Formula

 

 

Size

Side a

Side b

Side c

Perimeter

Ratio

s

s-a

s-b

s-c

Area

Ratio

1

1.7

2.3

3.9

7.9

 

3.95

2.25

1.65

0.05

0.86

 

2

3.4

4.6

7.8

15.8

2

7.9

4.5

3.3

0.1

3.43

4

3

5.1

6.9

11.7

23.7

3

11.85

6.75

4.95

0.15

7.71

9

4

6.8

9.2

15.6

31.6

4

15.8

9

6.6

0.2

13.70

16

5

8.5

11.5

19.5

39.5

5

19.75

11.25

8.25

0.25

21.41

25

6

10.2

13.8

23.4

47.4

6

23.7

13.5

9.9

0.3

30.83

36

7

11.9

16.1

27.3

55.3

7

27.65

15.75

11.55

0.35

41.96

49

8

13.6

18.4

31.2

63.2

8

31.6

18

13.2

0.4

54.80

64

9

15.3

20.7

35.1

71.1

9

35.55

20.25

14.85

0.45

69.36

81

10

17

23

39

79

10

39.5

22.5

16.5

0.5

85.63

100

11

18.7

25.3

42.9

86.9

11

43.45

24.75

18.15

0.55

103.61

121

12

20.4

27.6

46.8

94.8

12

47.4

27

19.8

0.6

123.30

144

13

22.1

29.9

50.7

102.7

13

51.35

29.25

21.45

0.65

144.71

169

14

23.8

32.2

54.6

110.6

14

55.3

31.5

23.1

0.7

167.83

196

15

25.5

34.5

58.5

118.5

15

59.25

33.75

24.75

0.75

192.66

225

16

27.2

36.8

62.4

126.4

16

63.2

36

26.4

0.8

219.21

256

17

28.9

39.1

66.3

134.3

17

67.15

38.25

28.05

0.85

247.47

289

18

30.6

41.4

70.2

142.2

18

71.1

40.5

29.7

0.9

277.44

324

19

32.3

43.7

74.1

150.1

19

75.05

42.75

31.35

0.95

309.12

361

20

34

46

78

158

20

79

45

33

1

342.51

400

 

 

Click here to open the Excel spreadsheet.  You can type in different values for the original side lengths and investigate each computation and graph for yourself.


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