# [[Surface Area and Volume of a Sphere]] (Mathematics > Geometry)
## Definition
The surface area and volume of a sphere are directly related to its radius $r$, defined as follows:
- **Surface Area**: The surface area $A$ of a sphere is the total area covered by its surface, given by:
$\large A = 4\pi r^2$
- **Volume**: The volume $V$ of a sphere is the total space within the sphere, defined as:
$\large V = \frac{4}{3}\pi r^3$
## Key Concepts
- Surface area is proportional to the square of the radius.
- Volume is proportional to the cube of the radius.
- Both formulas depend on the constant $\pi$.
## Important Properties
1. As the radius increases, surface area grows proportionally to the square of the radius.
2. As the radius increases, volume grows proportionally to the cube of the radius.
3. The volume formula can be derived from the surface area formula using integration.
## Essential Formulas
- Surface Area Formula:
$\large A = 4\pi r^2$
- Volume Formula:
$\large V = \frac{4}{3}\pi r^3$
## Core Examples
1. **Surface Area Calculation**: For a sphere with radius $r = 3$, the surface area is:
$A = 4 \pi (3)^2 = 36 \pi$
2. **Volume Calculation**: For a sphere with radius $r = 3$, the volume is:
$V = \frac{4}{3} \pi (3)^3 = 36 \pi$
## Related Theorems/Rules
- [[Sphere Volume Formula]]
- [[Volume Formulas in Higher Dimensions]]
## Common Pitfalls
- Forgetting the $\frac{4}{3}$ coefficient in the volume formula.
- Confusing the surface area and volume formulas.
## Related Topics
- [[Power Rule]]
- [[Area of a Triangle Using Trigonometry]]
- [[Integral of Power Functions with Linear Arguments]]
- [[Power Rule for Integration]]
- [[Rigid Motions]]
- [[Second Derivative]]
- [[Antiderivatives]]
- [[Indefinite Integrals]]
- [[Rotational Symmetry of Regular Polygons]]
- [[Circle Area]]
- [[Integral Calculus in Solid Geometry]]
## Quick Review Questions
1. What is the formula for the surface area of a sphere with radius $r$?
2. If the surface area of a sphere is $100 \pi$, what is its radius $r$?
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![[Surface_Area_and_Volume_of_a_Sphere_visualization]]