Question

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

Answer 1

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

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

$d=2\,r$

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

$\text{We know that,}$

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

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

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

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

$\text{We know that,}$

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

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

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

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

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