MainFragger
New member
- Joined
- Apr 19, 2017
- Messages
- 2
When I was in school many years ago, I remember learning about right triangles and Pythagorean Theorem. At one point the teacher pointed out that there is a bit of a simple shortcut to finding the hypoteneuse when side a and side b of the triangle are the same. Which is, you just use the number and the square root of two. In other words, if each side is 3, then its 3*sqrt(2)....if both sides are 4, then its 4*sqrt(2)..
Well, what happens when the sides are NOT equal? Initially, I assumed that whatever number you get, you divide by the sqrt(2) and you would be left with the difference between the two numbers you are using. Or the quotient. Or some variation of that...something that makes sense.. but when I use excel sheets to figure that out, well, lets just say that I haven't spotted it yet. I kind of toyed with the idea that if you subtract the highest possible square from the number, you would be left with a ratio that equates to how many line segments there are, versus how many are possible in the next triangle with matching sides. For example, with sides 3 and 4, the next triangle up would be 4 and 4, which is a total of 8 segments. So a triangle with 3 and 4 segments has 7/8 segments..
I thought maybe it would be more obvious if I pushed it so that some values were two numbers apart instead of one, but I still don't really see the pattern I am looking for.
I figure maybe someone more accomplished at math will see something that I might not notice.
The chart below (oops, just noticed a slight error in my first few cells, but don't worry, it doesn't affect the rest of the sheet) shows the sides in the first two columns, the length of the hypotenuse in the third column, and those values divided by the sqrt(2) in the fourth column).
Well, what happens when the sides are NOT equal? Initially, I assumed that whatever number you get, you divide by the sqrt(2) and you would be left with the difference between the two numbers you are using. Or the quotient. Or some variation of that...something that makes sense.. but when I use excel sheets to figure that out, well, lets just say that I haven't spotted it yet. I kind of toyed with the idea that if you subtract the highest possible square from the number, you would be left with a ratio that equates to how many line segments there are, versus how many are possible in the next triangle with matching sides. For example, with sides 3 and 4, the next triangle up would be 4 and 4, which is a total of 8 segments. So a triangle with 3 and 4 segments has 7/8 segments..
I thought maybe it would be more obvious if I pushed it so that some values were two numbers apart instead of one, but I still don't really see the pattern I am looking for.
I figure maybe someone more accomplished at math will see something that I might not notice.
The chart below (oops, just noticed a slight error in my first few cells, but don't worry, it doesn't affect the rest of the sheet) shows the sides in the first two columns, the length of the hypotenuse in the third column, and those values divided by the sqrt(2) in the fourth column).
1 | 2 | FALSE | #VALUE! | 1 | 3 | 3.162278 | 2.236068 |
2 | 3 | 3.605551 | 2.5495097567964 | 2 | 4 | 4.472136 | 3.162278 |
3 | 4 | 5 | 3.53553390593275 | 3 | 5 | 5.830952 | 4.123106 |
4 | 5 | 6.403124 | 4.52769256906872 | 4 | 6 | 7.211103 | 5.09902 |
5 | 6 | 7.81025 | 5.52268050859365 | 5 | 7 | 8.602325 | 6.082763 |
6 | 7 | 9.219544 | 6.51920240520267 | 6 | 8 | 10 | 7.071068 |
7 | 8 | 10.63015 | 7.51664818918648 | 7 | 9 | 11.40175 | 8.062258 |
8 | 9 | 12.04159 | 8.51469318296323 | 8 | 10 | 12.80625 | 9.055385 |
9 | 10 | 13.45362 | 9.51314879522026 | 9 | 11 | 14.21267 | 10.04988 |
10 | 11 | 14.86607 | 10.5118980208144 | 10 | 12 | 15.6205 | 11.04536 |
11 | 12 | 16.27882 | 11.5108644332214 | 11 | 13 | 17.02939 | 12.04159 |
12 | 13 | 17.69181 | 12.5099960031968 | 12 | 14 | 18.43909 | 13.0384 |
13 | 14 | 19.10497 | 13.5092560861063 | 13 | 15 | 19.84943 | 14.03567 |
14 | 15 | 20.51828 | 14.508618128547 | 14 | 16 | 21.26029 | 15.0333 |
15 | 16 | 21.93171 | 15.5080624192709 | 15 | 17 | 22.67157 | 16.03122 |
16 | 17 | 23.34524 | 16.5075740192192 | 16 | 18 | 24.08319 | 17.02939 |
17 | 18 | 24.75884 | 17.5071414000117 | 17 | 19 | 25.4951 | 18.02776 |
18 | 19 | 26.1725 | 18.5067555233218 | 18 | 20 | 26.90725 | 19.0263 |
19 | 20 | 27.58623 | 19.5064092031312 | 19 | 21 | 28.3196 | 20.02498 |
20 | 21 | 29 | 20.50609665441 | 20 | 22 | 29.73214 | 21.0238 |