Mathematical Inversion of the Sphere Volume Formula
When you know the volume \(V\) and must determine the radius \(r\), the standard Sphere Volume Formula \(V = \frac{4}{3}\pi r^3\) is inverted through algebra:
r = ³√(&frac{3V}{4π})
1. Multiply both sides by 3: \( 3V = 4\pi r^3 \)
2. Divide both sides by \( 4\pi \): \( \frac{3V}{4\pi} = r^3 \)
3. Take the cube root: \( r = \sqrt[3]{\frac{3V}{4\pi}} \)
Finding Radius from Other Parameters
Depending on the given information, use these formulas to calculate radius:
- From Diameter: \( r = \frac{d}{2} \)
- From Surface Area: \( r = \sqrt{\frac{A}{4\pi}} \)
- From Circumference: \( r = \frac{C}{2\pi} \)