Hello my friends. I wanted to know is it possible to calculate accurate circumference of a circle without its mathematical equations??? I mean using pure geometry???
Without any calculations at all? The only way I could think of is to put a tape measure around the circle!
Perhaps what you are thinking about is what Archimede's did. Imagine placing "n" points equally spaced around the circumference of the circle, draw lines between neighboring points The total lengths of those lines approximates the circumference of the circle and the larger you take n, the better the approximation is.
Of course, to get any sort of numerical answer, you will need draw lines from those points to the center of the circle so you get n triangles and then use some kind of formulas to relate those lengths to the vertex angle. Archimedes reduced those to square roots and then used a complicated scheme for approximating square roots to any accuracy to show that the circumference is a multiple of the diameter and then get an accurate value for that multiple, basically determining that pi is between 3 and 10/71 and 3 and 1/7. Here is a link to that:
Archimedes' Approximation of Pi
But the basic difficulty is your distinction between "mathematical formulas" and "pure geometry". The ratio of circumference to diameter of a circle is a number, not a geometric object. Whatever method you use, the result will be a "mathematical formula".
In 1685 Adam Adamandy Kochanski published a approximate construction of the transcendent number $\displaystyle \pi$ :
Adam Adamandy Kocha?ski - Wikipedia, the free encyclopedia
For the actual construction see here:
Näherungskonstruktion von Kocha?ski ? Wikipedia
Unfortunately this site is only available in German or Polish
The link is in German, but the construction is almost trivial to follow uust from the diagram. The red line (AZ) is just about equal to r time pi, half the circumference of the circle. And since it is a unit circle, AZ is pi to 5 significant figures.
The site also seems to show an excellent approximation to squaring the circle with a compas and straightedge.