To find the radius [tex]\( r \)[/tex] of a sphere given the surface area [tex]\( S \)[/tex] using the equation [tex]\( S = 4 \pi r^2 \)[/tex]:
1. Start with the surface area formula for a sphere:
[tex]\[
S = 4 \pi r^2
\][/tex]
2. To isolate [tex]\( r \)[/tex], we must first solve for [tex]\( r^2 \)[/tex]. Divide both sides of the equation by [tex]\( 4\pi \)[/tex]:
[tex]\[
\frac{S}{4 \pi} = r^2
\][/tex]
3. Next, to solve for [tex]\( r \)[/tex], take the square root of both sides. Since a square root can be positive or negative, we must consider both solutions:
[tex]\[
r = \pm \sqrt{\frac{S}{4 \pi}}
\][/tex]
4. Simplify the expression under the square root:
[tex]\[
r = \pm \frac{\sqrt{S}}{\sqrt{4 \pi}}
\][/tex]
5. Simplify the denominator further, knowing that [tex]\(\sqrt{4} = 2\)[/tex] and [tex]\(\sqrt{\pi} = \sqrt{\pi}\)[/tex]:
[tex]\[
r = \pm \frac{\sqrt{S}}{2 \sqrt{\pi}}
\][/tex]
Hence, the radius [tex]\( r \)[/tex] of the sphere given the surface area [tex]\( S \)[/tex] is:
[tex]\[
r = \pm \frac{\sqrt{S}}{2 \sqrt{\pi}}
\][/tex]
