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Math Help - square circle

  1. #1
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    square circle

    The perimeter of a semi-circle is same as the perimeter of a square . Than whether the area of the semi-circle is greater than area of square or vice versa????
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  2. #2
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    Hello, saroja!

    The perimeter of a semicircle is same as the perimeter of a square.
    Then: .[area of semi-circle] .(> or <) [area of square] ?
    A semicircle of radius r has a perimeter of \pi r + 2r
    A square of side x has a perimeter of 4x
    . . Hence, we have: . (\pi+ 2)r \;=\;4s\quad\Rightarrow\quad \frac{r}{s} \:=\:\frac{4}{\pi+2}

    Square both sides: . \frac{r^2}{x^2} \;=\;\frac{16}{(\pi+2)^2}


    Multiply both sides by \frac{1}{2}\pi\!:\;\;\frac{\frac{1}{2}\pi r^2}{x^2} \;=\;\frac{8\pi}{(\pi+2)^2}

    We have: . \frac{\text{Area of semicircle}}{\text{Area of square}} \;=\;\frac{8\pi}{(\pi+2)^2} \;\approx\;0.95


    Therefore: . \text{[Area of semicircle]} \;\;\bf{{\color{red}<}} \;\; \text{[Area of square]}

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  3. #3
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    Hello Mr.Soroban ,

    Thank you soo much ...
    I confused in calc perimeter of semi circle....

    Thank Q soo much for the clear explanation....
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