Mathematical Derivation & Proof: Formula for Volume of a Sphere
The mathematical equation for the volume of a sphere, V = ⁴⁄₃πr³, is one of the most elegant achievements in classical and modern mathematics.
🏛️ Archimedes' Historic Cylinder Theorem:
In 225 BC, Archimedes proved that a sphere inscribed inside a cylinder has exactly two-thirds the volume of that cylinder. Since the cylinder's volume is \( V_{ ext{cyl}} = ext{base} imes ext{height} = (\pi r^2)(2r) = 2\pi r^3 \), taking two-thirds yields: \[ V_{ ext{sphere}} = rac{2}{3} imes (2\pi r^3) = rac{4}{3}\pi r^3 \]
Calculus Proof via Disk Slices
We can also prove the formula by integrating thin circular cross-sectional disks along the horizontal axis from \( x = -r \) to \( x = +r \):
\[ V = \int_{-r}^{r} \pi y^2 \, dx = \int_{-r}^{r} \pi (r^2 - x^2) \, dx \]
\[ V = 2\pi \left[ r^2 x - rac{x^3}{3}
ight]_{0}^{r} = 2\pi \left( r^3 - rac{r^3}{3}
ight) = 2\pi \left( rac{2}{3}r^3
ight) = rac{4}{3}\pi r^3 \]