When I was young, I thought I could derive the value of pi as follows. Imagine a badly drawn hexagon as below. Every hexagon is the combination of 12 right angled triangles like the one shown in red.
The following is known from trigonometry.
If we extrapolate from a hexagon to a polygon of N sides, we get the following.
Note that 2Nx/2r is the ratio of the hexagon’s perimeter to its ‘radius’. As N approaches infinity, the polygon approaches a circle and the value of 2Nx/2r should approach pi.
Using the same principle, you can also easily derive tangent and cosine versions of the formulas to get the following.
While these equations work (you can test different values of N for yourself using a calculator), this is an invalid way of deriving pi. Trigonometrical functions are derived from pi and thus one cannot use them to derive pi itself. If it isn’t clear how the sine function and pi are related, the following animation (by Lucas V Barbarosa) is a good illustration.