archimedes pi

Question

Mar 1, 2008, 01:49 PM

caraa

New Member

Join Date: Mar 2008

Posts: 1

archimedes pi

For my maths gsce coursework I have to write about how archimedes estimated the value of pi. I have completed all of it up to an a grade. to get an a star I have to give this in terms of n. im really stuck can anyone help??

Answers

Mar 1, 2008, 02:11 PM	#2

galactus

Mathematics Expert

Join Date: Sep 2006

Location: Chaneysville, Pa.

Posts: 939

There is lots and lots on the web about this. Why not just google it?.

It is rather extensive to show all the details,

Looking at fig. 3, we have $(1-x)^{2}+(s/2)^{2}=1........[1]$

and $x^{2}+(s/2)^{2}=t^{2}...........[2]$

Squaring [1] gives us $(1-2x+x^{2}+\frac{s^{2}}{4}=1$

or $-2x+(x^{2}+\frac{s^{2}}{4})=0$

Now use [2] to make a substitution:

$-2x+t^{2}=0$ or $x=\frac{t^{2}}{2}$

On the other hand, [1] yields us $(1-x)^{2}=1-\frac{s^{2}}{4}=\frac{1}{4}(4-s^{2})$

Square root of both sides:

$1-x=\frac{1}{2}\sqrt{4-s^{2}}$

$1-\frac{1}{2}\sqrt{4-s^{2}}=x=\frac{t^{2}}{2}$

or $\frac{t^{2}}{2}=1-\frac{1}{2}\sqrt{4-s^{2}}$

Therefore, we multiply by 2 and get:

$t^{2}=2-\sqrt{4-s^{2}}$ or $t=\sqrt{2\sqrt{4-s^{2}}}$

Finally, with $t=s_{n+1}, \;\ and \;\ s=s_{n}$ ,

we get a formula for the side of the (n+1)st polygon in terms of that of the nth polygon:

$s_{n+1}=\sqrt{2-\sqrt{4-s_{n}^{2}}}$

$s_{1}$ is the side of the inscribed square , so we get:

$s_{1}=\sqrt{2}$

Double the sides and get 8 sides:

$s_{2}=\sqrt{2-\sqrt{2}}$

Double it and get 16 sides:

$s_{3}=\sqrt{2-\sqrt{2+\sqrt{2}}}$ .

In general,

$s_{n}=\sqrt{2-\sqrt{2+\sqrt{2+.....\sqrt{2+\sqrt{2}}}}}$

Since the perimeter of the nth polygon is $p_{n}=2^{n+1}$ , an

approximation to $\pi$ is $\frac{1}{2}p_{n}=2^{n}s_{n}$

or

${\pi}_{n}=2^{n}\sqrt{2-\sqrt{2+\sqrt{2+....\sqrt{2+\sqrt{2}}}}}$ , where we have n radicals.

This can be written in a better way as ${\pi}_{n}=2^{n}\sqrt{2-2_{n-1}}$

For instance, if we use n=8 in the above formula, we get 3.14157294025

I believe Archimedes used up to a 96-sided polygon.

I hope this helps. Remember, Archimedes did not use trig as we know it.

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