Prove that the surface area of a sphere of diameter d is pi*d2 and the volume is 1/6*pi*d3

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Prove that the surface area of a sphere of diameter d is pi*d2 and the volume is 1/6*pi*d3

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Hailey 1 month 2021-08-18T08:28:23+00:00 1 Answer 0 views 0

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    2021-08-18T08:29:27+00:00

    \text{Let 'r' be the radius of the sphere}

    \text{Since 'd' is the diameter, we have}

    d=2\,r

    \implies\,r=\dfrac{d}{2}

    \text{We know that,}

    \textbf{Surface area of sphere}\bf\;=4\,\pi\,r^2

    \text{Surface area of sphere}=4\,\pi\,(\dfrac{d}{2})^2

    \text{Surface area of sphere}=4\,\pi\,\dfrac{d^2}{4}

    \implies\boxed{\textbf{Surface area of sphere}\bf=\pi\,d^2}

    \text{We know that,}

    \textbf{Volume of sphere}\bf\;=\dfrac{4}{3}\,\pi\,r^3

    \text{Volume of sphere}=\dfrac{4}{3}\,\pi\,(\dfrac{d}{2})^3

    \text{Volume of sphere}=\dfrac{4}{3}\,\pi\,\dfrac{d^3}{8}

    \text{Volume of sphere}=\dfrac{1}{3}\,\pi\,\dfrac{d^3}{2}

    \implies\boxed{\textbf{Volume of sphere}\bf=\dfrac{1}{6}\pi\,d^3}

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