osanseviero one year ago Calculating an approximation of pi using a square inside a circle

1. osanseviero

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2. osanseviero

I need to find a close approximation of pi using a square inside a circle

3. freckles

|dw:1441563708086:dw| well you could say $\text{ small square area }<\text{ circle area } < \text{ big square area }$ |dw:1441563787054:dw| |dw:1441563837206:dw| so you have: $r^2 <\text{ circle area} <R^2 \\ \text{ hmmm... circle area} \approx \frac{R^2+r^2}{2}$

4. osanseviero

Ok. I also know that the radius is 1. Let me calculate the areas

5. freckles

well I didn't know if we were just limited to using the square inside the circle..

6. osanseviero

Well..how would you do it with two squares?

7. freckles

I drew two squares above..

8. freckles

what do you mean?

9. osanseviero

Sorry, I explained myself wrong. Give me a minute, please

10. freckles

oops and I wrote things wrong aboe

11. freckles

$(2r)^2< \text{ circle area } < (2R^2) \\ \text{ circle area } \approx \frac{(2r)^2+(2R)^2}{2}$

12. freckles

|dw:1441564320576:dw||dw:1441564336095:dw|

13. freckles

I used R and r as half the side lengths above

14. osanseviero

Yep. So we know that R is 1, which should make things a little easier. How can we calculate r?

15. osanseviero

Right now I have pi = 2r^2 + 2

16. freckles

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17. freckles

r is sqrt(2)/2

18. freckles

$2(\frac{\sqrt{2}}{2})^2+2 =2(\frac{2}{4})+2=1+2=3$

19. osanseviero

Great! Thanks a lot. And any idea of how to do this with 1 square? (just for curiosity)

20. freckles

well I guess you can get a lower approximation of the circle just by consider the area of the square

21. freckles

area of the small square*

22. osanseviero

Ok, thanks a lot :) Have a nice day

23. freckles

you too

24. freckles

it kind takes me back to calculus days

25. freckles

we can find the lower sum and the upper sum but I think the average of those sums gave a better approximation

26. osanseviero

They told me something about finding the max square that could be inside the circle

27. freckles

we can find the lower sum and the upper sum but I think the average of those sums gave a better approximation

28. osanseviero

Ok,thanks again :)