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.
Return to Triangle Investigations
Return to my EMAT 6690 Homepage