Example 1
\(r = 3 \rightarrow V = \frac{4}{3}\pi(3^3) = \frac{4}{3}\pi(27) \approx 113.10\) units³, \(SA = 4\pi(3^2) \approx 113.10\) units²
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Theory & Formula
A sphere is a perfectly round three-dimensional shape where every point on the surface is equidistant from the center. This distance is the radius.
The volume represents the amount of space inside the sphere, while the surface area is the total area covering the outside of the sphere. Both formulas depend only on the radius.
Properties and Applications
- A sphere has the smallest surface area for a given volume among all 3D shapes
- All cross-sections through the center are circles of radius r
- The diameter is twice the radius: d = 2r
- Used to calculate volumes of planets, balls, bubbles, and other spherical objects
- Relationship: Volume of sphere = 2/3 × Volume of circumscribing cylinder
\(V = \frac{4}{3}\pi r^3, SA = 4\pi r^2\)
Worked Examples
Example 2
\(r = 5 \rightarrow V = \frac{4}{3}\pi(5^3) = \frac{4}{3}\pi(125) \approx 523.60\) units³, \(SA = 4\pi(5^2) \approx 314.16\) units²