how to derive relation between solid angle and surface area and the radius of sphere ?
I know $s=r^2\Omega$ but how they got it using integral ?
2026-03-13 15:04:42.1773414282
how to derive relation between solid angle and surface area and the radius of sphere using definite integral?
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One way is to start from the volume of a unit sphere. Using the disk method, it is $\int_{-1}^1 \pi (1-z^2)dz=\pi(z-\frac{z^3}3)|_{-1}^1=\frac 43\pi$ The volume of a sphere of radius $r$ is then $\frac 43 \pi r^3$. The surface area is then the derivative of this with respect to $r$-think of increasing $r$ a small amount $dr$ and the volume added. This gives the total solid angle is $\frac d{dr} \frac 43 \pi r^3=4\pi r^2$