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 A better approximate Pythagorean Theorem (Posted on 2017-02-08)
Adding half the shorter leg to the longer leg of a right triangle does give a pretty good approximation of the hypotenuse of a right triangle but it is always an overestimate.

Reduce the half to a value where the maximum % error (larger or smaller) is minimized.

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 heuristic estimate based on computer program Comment 1 of 1
0.336361

based on trial-and-error in

DefDbl A-Z
Dim crlf\$

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For longer = 1 To 10 Step 0.1
DoEvents
hyp = Sqr(1 + longer * longer)
approx = 0.336361 + longer
Text1.Text = Text1.Text & mform(longer, " #0.0000")
Text1.Text = Text1.Text & mform(hyp, " ##0.0000")
Text1.Text = Text1.Text & mform(approx, " ##0.0000")
Text1.Text = Text1.Text & mform(approx - hyp, " ##0.0000")
Text1.Text = Text1.Text & mform((approx - hyp) / hyp, " ##0.000000")
Text1.Text = Text1.Text & crlf
Next longer

Text1.Text = Text1.Text & ct & crlf & " done"

End Sub

Function mform\$(x, t\$)
a\$ = Format\$(x, t\$)
If Len(a\$) < Len(t\$) Then a\$ = Space\$(Len(t\$) - Len(a\$)) & a\$
mform\$ = a\$
End Function

where the results of this final stage are:

```   leg                                relative
ratio     true    approx    diff      error
1.0000   1.4142   1.3364 - 0.0779 - 0.055050  1.1000   1.4866   1.4364 - 0.0502 - 0.033799  1.2000   1.5620   1.5364 - 0.0257 - 0.016446  1.3000   1.6401   1.6364 - 0.0038 - 0.002293  1.4000   1.7205   1.7364   0.0159   0.009239  1.5000   1.8028   1.8364   0.0336   0.018630  1.6000   1.8868   1.9364   0.0496   0.026269  1.7000   1.9723   2.0364   0.0641   0.032476  1.8000   2.0591   2.1364   0.0772   0.037509  1.9000   2.1471   2.2364   0.0893   0.041577  2.0000   2.2361   2.3364   0.1003   0.044852  2.1000   2.3259   2.4364   0.1104   0.047473  2.2000   2.4166   2.5364   0.1198   0.049554  2.3000   2.5080   2.6364   0.1284   0.051186  2.4000   2.6000   2.7364   0.1364   0.052447  2.5000   2.6926   2.8364   0.1438   0.053398  2.6000   2.7857   2.9364   0.1507   0.054092  2.7000   2.8792   3.0364   0.1571   0.054572  2.8000   2.9732   3.1364   0.1631   0.054872  2.9000   3.0676   3.2364   0.1688   0.055024  3.0000   3.1623   3.3364   0.1741   0.055050  3.1000   3.2573   3.4364   0.1791   0.054972  3.2000   3.3526   3.5364   0.1838   0.054808  3.3000   3.4482   3.6364   0.1882   0.054572  3.4000   3.5440   3.7364   0.1924   0.054275  3.5000   3.6401   3.8364   0.1963   0.053929  3.6000   3.7363   3.9364   0.2001   0.053543  3.7000   3.8328   4.0364   0.2036   0.053123  3.8000   3.9294   4.1364   0.2070   0.052676  3.9000   4.0262   4.2364   0.2102   0.052208  4.0000   4.1231   4.3364   0.2133   0.051722  4.1000   4.2202   4.4364   0.2162   0.051223  4.2000   4.3174   4.5364   0.2190   0.050714  4.3000   4.4147   4.6364   0.2216   0.050198  4.4000   4.5122   4.7364   0.2242   0.049678  4.5000   4.6098   4.8364   0.2266   0.049154  4.6000   4.7074   4.9364   0.2289   0.048629  4.7000   4.8052   5.0364   0.2312   0.048105  4.8000   4.9031   5.1364   0.2333   0.047583  4.9000   5.0010   5.2364   0.2354   0.047063  5.0000   5.0990   5.3364   0.2373   0.046546  5.1000   5.1971   5.4364   0.2392   0.046034  5.2000   5.2953   5.5364   0.2411   0.045527  5.3000   5.3935   5.6364   0.2428   0.045026  5.4000   5.4918   5.7364   0.2445   0.044530  5.5000   5.5902   5.8364   0.2462   0.044040  5.6000   5.6886   5.9364   0.2478   0.043557  5.7000   5.7871   6.0364   0.2493   0.043080  5.8000   5.8856   6.1364   0.2508   0.042610  5.9000   5.9841   6.2364   0.2522   0.042147  6.0000   6.0828   6.3364   0.2536   0.041691  6.1000   6.1814   6.4364   0.2549   0.041242  6.2000   6.2801   6.5364   0.2562   0.040801  6.3000   6.3789   6.6364   0.2575   0.040366  6.4000   6.4777   6.7364   0.2587   0.039938  6.5000   6.5765   6.8364   0.2599   0.039518  6.6000   6.6753   6.9364   0.2610   0.039104  6.7000   6.7742   7.0364   0.2621   0.038697  6.8000   6.8731   7.1364   0.2632   0.038298  6.9000   6.9721   7.2364   0.2643   0.037905  7.0000   7.0711   7.3364   0.2653   0.037518  7.1000   7.1701   7.4364   0.2663   0.037138  7.2000   7.2691   7.5364   0.2672   0.036765  7.3000   7.3682   7.6364   0.2682   0.036398  7.4000   7.4673   7.7364   0.2691   0.036037  7.5000   7.5664   7.8364   0.2700   0.035683  7.6000   7.6655   7.9364   0.2709   0.035334  7.7000   7.7647   8.0364   0.2717   0.034992  7.8000   7.8638   8.1364   0.2725   0.034655  7.9000   7.9630   8.2364   0.2733   0.034324  8.0000   8.0623   8.3364   0.2741   0.033998  8.1000   8.1615   8.4364   0.2749   0.033678  8.2000   8.2608   8.5364   0.2756   0.033364  8.3000   8.3600   8.6364   0.2763   0.033055  8.4000   8.4593   8.7364   0.2770   0.032750  8.5000   8.5586   8.8364   0.2777   0.032451  8.6000   8.6579   8.9364   0.2784   0.032157  8.7000   8.7573   9.0364   0.2791   0.031868  8.8000   8.8566   9.1364   0.2797   0.031584  8.9000   8.9560   9.2364   0.2804   0.031304  9.0000   9.0554   9.3364   0.2810   0.031029  9.1000   9.1548   9.4364   0.2816   0.030758  9.2000   9.2542   9.5364   0.2822   0.030491  9.3000   9.3536   9.6364   0.2828   0.030229  9.4000   9.4530   9.7364   0.2833   0.029971  9.5000   9.5525   9.8364   0.2839   0.029717  9.6000   9.6519   9.9364   0.2844   0.029467  9.7000   9.7514  10.0364   0.2850   0.029222  9.8000   9.8509  10.1364   0.2855   0.028979  9.9000   9.9504  10.2364   0.2860   0.028741 10.0000  10.0499  10.3364   0.2865   0.028506```

 Posted by Charlie on 2017-02-08 10:51:42

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