圓周率怎樣計出來？

Use calculus to find a "related value" of pi, say (pi/4)
then use the Riemann Sum or other expansion to find the numerical value
if you add more terms up~ you get higher accuracy
these days nobody measure the circle. i guess you agree with me

but the key points now are
1)how fast the expansion converges to pi
2)is it difficult to find one term in the expansion (if it is difficult, you need more time to add up so many terms)
3)how many significant figures can the calculating machines hold. eg, your calculator most likely cannot hold more that 16 decimal places. so you can make use of an expansion and your calculator to find pi, but you never get an accuracy higher than 16 d.p.


Leibniz's series was very famous in the past
but it has very low rate of convergence
(the form is so simple!!)
later, people found out more complicated series which converges to pi faster

http://en.wikipedia.org/wiki/Leibniz_formula_for_pi
http://www.mathpath.net/Leibniz.htm

去看維基百科吧，超多東西看。