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osanseviero

  • one year ago

Calculating an approximation of pi using a square inside a circle

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  1. osanseviero
    • one year ago
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    |dw:1441563540851:dw|

  2. osanseviero
    • one year ago
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    I need to find a close approximation of pi using a square inside a circle

  3. freckles
    • one year ago
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    |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
    • one year ago
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    Ok. I also know that the radius is 1. Let me calculate the areas

  5. freckles
    • one year ago
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    well I didn't know if we were just limited to using the square inside the circle..

  6. osanseviero
    • one year ago
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    Well..how would you do it with two squares?

  7. freckles
    • one year ago
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    I drew two squares above..

  8. freckles
    • one year ago
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    what do you mean?

  9. osanseviero
    • one year ago
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    Sorry, I explained myself wrong. Give me a minute, please

  10. freckles
    • one year ago
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    oops and I wrote things wrong aboe

  11. freckles
    • one year ago
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    \[(2r)^2< \text{ circle area } < (2R^2) \\ \text{ circle area } \approx \frac{(2r)^2+(2R)^2}{2}\]

  12. freckles
    • one year ago
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    |dw:1441564320576:dw||dw:1441564336095:dw|

  13. freckles
    • one year ago
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    I used R and r as half the side lengths above

  14. osanseviero
    • one year ago
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    Yep. So we know that R is 1, which should make things a little easier. How can we calculate r?

  15. osanseviero
    • one year ago
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    Right now I have pi = 2r^2 + 2

  16. freckles
    • one year ago
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    |dw:1441564624855:dw|

  17. freckles
    • one year ago
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    r is sqrt(2)/2

  18. freckles
    • one year ago
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    \[2(\frac{\sqrt{2}}{2})^2+2 =2(\frac{2}{4})+2=1+2=3\]

  19. osanseviero
    • one year ago
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    Great! Thanks a lot. And any idea of how to do this with 1 square? (just for curiosity)

  20. freckles
    • one year ago
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    well I guess you can get a lower approximation of the circle just by consider the area of the square

  21. freckles
    • one year ago
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    area of the small square*

  22. osanseviero
    • one year ago
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    Ok, thanks a lot :) Have a nice day

  23. freckles
    • one year ago
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    you too

  24. freckles
    • one year ago
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    it kind takes me back to calculus days

  25. freckles
    • one year ago
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    we can find the lower sum and the upper sum but I think the average of those sums gave a better approximation

  26. osanseviero
    • one year ago
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    They told me something about finding the max square that could be inside the circle

  27. freckles
    • one year ago
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    we can find the lower sum and the upper sum but I think the average of those sums gave a better approximation

  28. osanseviero
    • one year ago
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    Ok,thanks again :)

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