Biggie Size it

Investigation using

Microsoft Excel

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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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