Value of Pi derivation

Question

Derive the value of Pi. I want with explanation. Is there any possible way?

How do scientists calculate it?

Answer 1

This is probably not the standard way at all but one possible way to derive the value for $ \pi $ is to consider the following diagram.

We know the circumference of the circle is $2\pi r$ so we set $|OA|= 1$

Now we get a poor approximation to by saying it is: $2 \cdot |AB| = 2 \cdot \sqrt{2} \approx 2.82$

If we bisect AB at E and draw a segment from O through E to the Circumference at C we can get a better estimate of $\pi$ by saying $\pi = 4 \cdot |BC|$

We know $\angle OEB $ is a right angle so can use Pythagoras to find $|OE|$

$|OE| = \sqrt{1^2-|EB|^2}$ and $\dfrac{\sqrt{2}}{2}$ so $|OE| = \sqrt{1^2 - \left(\dfrac{\sqrt{2}}{2}\right)^2} = \sqrt{1 - \dfrac{1}{2}} \approx 0.707106$

From this we see $|EC| = 1-|OE| \approx 0.292894$ and we get

$|BC| = \sqrt{|EC|^2+|EB|^2} \approx\sqrt{0.292894^2 + 0.707106^2} \approx 0.765333$

And a better approximation to $\pi$ is $4 \cdot |BC| \approx 3.061332$

Now you can get an even better approximation to $\pi$ by saying $\pi = 8 \cdot |BD|$

We can get bisecting BC at F producing OF to D. I wont do the math but you can easily see that $|DB|$ is not difficult to find using similar arguments to the ones I used to find |BC| and you can keep bisecting to get better and better estimates of $\pi$