Surface Area of Spheres – SA = 4πr²
A sphere's surface area is given by the elegant formula SA = 4πr² – exactly 4 times the area of a circle with the same radius. This remarkable factor of 4 was proven by Archimedes over 2,000 years ago, and it remains one of the most beautiful results in all of geometry.
Because surface area depends on r², doubling a sphere's radius doesn't just double its surface area – it quadruples it. This squared relationship between radius and surface area is a pattern worth remembering, since it shows up throughout geometry wherever area (as opposed to length) is involved.
Calculating a Sphere's Surface Area
SA = 4πr². Because surface area depends on r², scaling the radius by a factor of k scales the surface area by k².
SA = 4π(6)² = 4π(36) = 144π ≈ 452.39 cm².
Surface area scales with r²: 3² = 9 times larger.
Real-Life Application
- Weather balloons: the fabric needed for a spherical balloon depends on its surface area.
- Planetary science: estimating a planet's surface area from its radius.
- Sports equipment: the material used to cover a ball.
Key Takeaways
- A sphere's surface area formula is SA = 4πr².
- It equals exactly 4 times the area of a circle with the same radius.
- Surface area scales with the square of the radius.
Practice: Surface Area of Spheres
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