in case you be conscious of 3 trigonometry, we are able to instruct that pi is below 22/7. Take a circle C of diameter a million. If P is a huge-unfold n-gon circumscribed approximately C, then the two aspects of P has length tangent(pi/n). hence the edge of P is n*tan(pi/n). because's circumscribed approximately C, the edge of P is larger than the edge of C, that's pi. So pi is below n*tan(pi/n). it somewhat is genuine for all integers n=3 or greater desirable. in case you put in n=ninety one you get that pi is below 3.142841..., that's below 22/7=3.142857... in specific pi isn't 22/7. Edit: To volter. this may be a stable element, that i did no longer instruct why the circumscribed perimeter could desire to be greater desirable. It somewhat gets right down to how one defines the size of a curve; Archimedes addressed it contained in right here way. First we define the thought of a curve being "concave" in a undeniable direction, in that secant lines lie on an identical edge of it. Now think that we've 2 curves with basic endpoints that are the two concave interior an identical direction. Then the two are on an identical edge of the at present line becoming a member of the factors. Now a line is the shortest distance between 2 factors, so intuitively the curve nearer to the line could desire to have shorter length than the different one. The nearer one is "nearer to being a line". In our occasion, we take as endpoints the n factors the place the polygon intersects the circle, and evaluate the arcs to the areas of the polygon. those are concave interior an identical direction and the polygon section is on the different area from the secant line. a community argument as you advise could artwork even with the shown fact that it provides a point of complexity (what's section? why is the component to the circle pi r squared?) opposite to my style for this concern.