Intermath:Investigations: Geometry: 3-D

In simplest terms, what is the ratio of the number of cubic inches in the volume of a cube to the number of square centimeters in the surface area of the cube, given that one edge is s centimeters long?

Look at the chart below to see how I discovered the answer through the use of patterns.

Sidle Length	Volume	Surface Area	Vol : Surface Area	Ratio/Side Length
1	1	6	6	6
2	8	24	3	6
3	27	54	2	6
4	64	96	1.5	6
5	125	150	1.2	6
6	216	216	1	6
7	343	294	0.857142857142857	6
8	512	384	0.75	6
9	729	486	0.666666666666667	6
10	1000	600	0.6	6
11	1331	726	0.545454545454545	6
12	1728	864	0.5	6
13	2197	1014	0.461538461538462	6
14	2744	1176	0.428571428571429	6
15	3375	1350	0.4	6
16	4096	1536	0.375	6
17	4913	1734	0.352941176470588	6
18	5832	1944	0.333333333333333	6
19	6859	2166	0.315789473684211	6
20	8000	2400	0.3	6
21	9261	2646	0.285714285714286	6
22	10648	2904	0.272727272727273	6
23	12167	3174	0.260869565217391	6
24	13824	3456	0.25	6
25	15625	3750	0.24	6
26	17576	4056	0.230769230769231	6
27	19683	4374	0.222222222222222	6
28	21952	4704	0.214285714285714	6
29	24389	5046	0.206896551724138	6
30	27000	5400	0.2	6
31	29791	5766	0.193548387096774	6
32	32768	6144	0.1875	6
33	35937	6534	0.181818181818182	6
34	39304	6936	0.176470588235294	6
35	42875	7350	0.171428571428571	6
36	46656	7776	0.166666666666667	6
37	50653	8214	0.162162162162162	6
38	54872	8664	0.157894736842105	6
39	59319	9126	0.153846153846154	6
40	64000	9600	0.15	6

The pattern that I discovered was that the ration of the volume to the surface area was equal to the ratio of the side length to six. So this means volume/surface area = side length/6. So in simplest terms, if the ratio of the volume to the surface area of a cube with side length s would be s/6.

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