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.