Volume of Spheres – V = &frac43;πr³
A sphere's volume is given by V = &frac43;πr³ – one of the most famous formulas in all of geometry. Archimedes discovered this formula (along with the sphere's surface area) more than 2,000 years ago, using a clever comparison between a sphere, a cylinder, and a cone, and considered it his proudest mathematical achievement.
Because the formula depends on r³ (the radius cubed), scaling a sphere's radius has a dramatic effect on its volume: doubling the radius doesn't double the volume, or even quadruple it – it multiplies the volume by 2³ = 8.
Calculating a Sphere's Volume
V = &frac43;πr³. Because volume depends on r³, scaling the radius by a factor of k scales the volume by k³.
V = &frac43;π(6)³ = &frac43;π(216) = 288π ≈ 904.78 cm³.
Volume scales with r³: 2³ = 8 times larger.
Real-Life Application
- Balloons: calculating how much gas fills a spherical balloon.
- Planetary science: estimating a planet's volume from its radius.
- Sports science: comparing the volume of balls of different sizes.
Key Takeaways
- A sphere's volume formula is V = &frac43;πr³.
- Archimedes proved this formula over 2,000 years ago and considered it his greatest achievement.
- Volume scales with the cube of the radius – a small radius change has a large volume effect.
Practice: Volume of Spheres
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