Understanding the Power of r³
In the formula V = 4/3 π r³, the radius is raised to the third power (cubed). This property leads to a fundamental principle of 3D geometry known as the Square-Cube Law.
The Scaling Multiplier
When you scale the radius of a sphere by a factor of k, the volume changes by k³:
- If you double the radius (k = 2), the volume increases by
2³ = 8 times. - If you triple the radius (k = 3), the volume increases by
3³ = 27 times. - If you halve the radius (k = 0.5), the volume drops to
(1/2)³ = 1/8thof the original volume.
Real-World ApplicationBall Bearings Scaling
A bearing with radius 2 mm has volume: V ≈ 33.51 mm³.
A bearing with radius 4 mm has volume: V ≈ 268.08 mm³.
Notice: 268.08 / 33.51 = 8.00!