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Math Help - Expression of Formula In Some Terms

  1. #1
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    Expression of Formula In Some Terms

    The formula for the area of a spherical cap with radius (the sphere's radius) R and height h is given as

    A = 2{\pi}Rh

    This formula can be expressed in terms of R and r, where r is the radius of the base of the spherical cap:

    (R-h)^2 + r^2 = R^2
    (R-h)^2 = R^2 - r^2
    R-h = \sqrt{R^2 - r^2}
    h = R - \sqrt{R^2 - r^2}

    A = 2{\pi}R(R - \sqrt{R^2 - r^2})

    Now, how do I express the formula in terms of R^2 and \frac{r}{R}? I've been trying for quite some time but I'm stuck...
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  2. #2
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    Quote Originally Posted by fishcake View Post

    This formula can be expressed in terms of R and r, where r is the radius of the base of the spherical cap:

    (R-h)^2 + r^2 = R^2
    (R-h)^2 = R^2 - r^2
    For r:

    (R-h)^2 = R^2 - r^2

    R^2-2Rh+h^2 = R^2 - r^2

    -2Rh+h^2 =  - r^2

    2Rh-h^2 =   r^2

    \sqrt{2Rh-h^2} =   r




    For R:

    (R-h)^2 = R^2 - r^2

    R^2-2Rh+h^2 = R^2 - r^2

    -2Rh+h^2 =  - r^2

    2Rh-h^2 =   r^2

    2Rh =   r^2+h^2

    R =  \frac{ r^2+h^2}{2h}
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  3. #3
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    Quote Originally Posted by fishcake View Post

    Now, how do I express the formula in terms of R^2 and \frac{r}{R}? I've been trying for quite some time but I'm stuck...


    For R^2:

     <br />
R = \frac{ r^2+h^2}{2h} \implies R^2 = \left(\frac{ r^2+h^2}{2h}\right)^2 =\dots<br />




    For \frac{r}{R}:

     \frac{r}{R} = \frac{\sqrt{2Rh-h^2}}{\frac{ r^2+h^2}{2h}} = \dots
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  4. #4
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    Hmm, I'm still confused here. How do I make use of \frac{r}{R} and R^2 inside the formula for A? What does "express A in terms of \frac{r}{R} and R^2" actually means anyway? Maybe I have misinterpreted the question, which can be found here. It's question 1(b).
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