I cut off one corner of an empty box to form an open 'pocket', so that each of the length a, breadth b, and depth c, of the initial corner was a different whole number of centimetres, and the area of each face of the 'pocket' was also an integer.
What is the area of the open 4th side of the 'pocket', in terms of a,b, and c?
(In reply to
re: what is wanted? by broll)
Yes indeed, area = sqrt(a^2 * b^2 + b^2 * c^2 + a^2 * c^2) / 2.
The second lines, added below the original lines, show the sum of the squares of the pairwise products of a,b,c, and then that divided by the square of the area. I knew the sum of the squares of the products was a fourth power of linear size, so I had to divide by the square of the area, and voila.
1 2 8 9.0000000000 11
324 4
2 4 6 14.0000000000 12
784 4
2 6 8 26.0000000000 16
2704 4
2 8 10 42.0000000000 20
7056 4
4 6 10 38.0000000000 20
5776 4
2 3 16 29.0000000000 21
3364 4
2 4 16 36.0000000000 22
5184 4
6 8 9 51.0000000000 23
10404 4
2 10 12 62.0000000000 24
15376 4
4 8 12 56.0000000000 24
12544 4
3 8 14 61.0000000000 25
14884 4
4 7 16 66.0000000000 27
17424 4
5 8 14 69.0000000000 27
19044 4
2 12 14 86.0000000000 28
29584 4
4 6 18 66.0000000000 28
17424 4
4 10 14 78.0000000000 28
24336 4
6 8 14 74.0000000000 28
21904 4
8 10 11 81.0000000000 29
26244 4
and it works regardless of whether the open side is integral:
1 2 4 4.5825756950 7
84 4
1 2 6 6.7823299831 9
184 4
2 3 4 7.8102496759 9
244 4
1 2 8 9.0000000000 11
324 4
1 4 6 12.5299640861 11
628 4
2 3 6 11.2249721603 11
504 4
2 4 5 11.8743420870 11
564 4
2 4 6 14.0000000000 12
784 4
1 2 10 11.2249721603 13
504 4
1 4 8 16.6132477258 13
1104 4
2 3 8 14.7309198627 13
868 4
2 4 7 16.1554944214 13
1044 4
2 5 6 16.9115345253 13
1144 4
3 4 6 16.1554944214 13
1044 4
2 4 8 18.3303027798 14
1344 4
1 2 12 13.4536240471 15
724 4
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Posted by Charlie
on 2014-08-15 10:07:19 |
re(2): what is wanted?
