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I asked the same question some time ago, but it is closed. This time I will be clearer (I hope).

If I have a right triangle $c ^ 2 = a ^ 2 + b ^ 2$, and $b > a$ (so $a$ is the shorter side and $b$ is the longer side). Now, put a right triangle in a semi-circle with the hypotenuse is equal to the diameter, $((a * 4) - (a * b / c)) / 3$ ~ circle arc $a$.

For example if have right triangle $10 ^ 2 = 5 ^ 2 + (8,6602\dots)^ 2$ which was placed in a semi-circle with a diameter of $10$, will obtain $((5 * 4) - (5 * 8,6602\dots / 10)) / 3 = 5,223\dots \sim 5,235\dots$.

In this example 5,235.. is the length of circle arc with the same angle as the right triangle side of 5, in this case it is obvious angle of 60 degrees (viewed from the center of the semi- circle).

Another example if have a right triangle $10 ^ 2 = (7,07106\dots)^ 2 + (7,07106\dots)^2$, will obtain $((7,07106\dots* 4) - (7,07106\dots* 7,07106\dots/ 10)) / 3 = 7,761 \sim 7,853\dots$. In this case $7,853\dots$ obvious is the length of 1/4 circumference, or, circle arc of 90 degrees (viewed from the center of the semi-circle).

So question is, whether formula $((a * 4) - (a * b / c)) / 3$, used before in this or some other form ? (question is important because this formula can be improved (with a lot of manipulation) to give a much smaller error) Srbin

EDIT (M.S): This picture was not posted by OP, I've added it since it might help to clarify the question.


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-1: Please make paragraphs, learn to use $\LaTeX$ and try to formulate proper sentences. Noone will enjoy reading this. –  Listing Nov 27 '11 at 13:25
marko: Your question is not very clear, I tried to improve the formatting. I've added the picture, could you explain it on this picture. Are you asking about the length of the arc between $B$ and $A$. (I.e. this length is $r.\phi$?) –  Martin Sleziak Nov 27 '11 at 14:15
P.S. I am not sure whether this is a good suggestion but there were several questions here posted in different languages than English and in these cases someone translated them. If you're lucky, there might be someone here who speaks your language and this could lead to a better question. –  Martin Sleziak Nov 27 '11 at 14:16
Now back to your question: If I understand you correctly, your formula gives $\frac 13 \left(4a-\frac{ab}c\right)= \frac13 r(8\tan(\phi/2)-\sin\phi)$. For small values of $\phi$ this is approximately $\frac13 r(4\phi-\phi)=\phi$. See wolframalpha.com/input/… and wolframalpha.com/input/… –  Martin Sleziak Nov 27 '11 at 14:18
Marko: I guess both these account belong to you: math.stackexchange.com/users/20189/marko and math.stackexchange.com/users/21380/marko; Maybe it would be good for you to register, so that you can better follow all questions you posted. (After you do it, you can even ask moderators to merge you account with the older unregistered ones.) –  Martin Sleziak Dec 17 '11 at 7:52

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