Sphere
Definition, properties, and formulas for sphere calculations
Definition and Properties
A sphere is a three-dimensional geometric body consisting of all points in space that have the same distance from one center point.
Core definitions:
- Radius \(r\): distance from center to surface
- Diameter \(d\): longest distance through the center, \(d = 2r\)
- Great circle: largest circular cross-section of the sphere
The surface area grows with \(r^2\), while the volume grows with \(r^3\).
This explains why volume increases much faster than surface when radius increases.
Formulas
Radius
Diameter
Surface area
Great-circle circumference
Cutting surface
Volume
\(\displaystyle d=2r\), \(\displaystyle A=4\pi r^2\), \(\displaystyle V=\frac{4}{3}\pi r^3\)
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