# Sphere

A sphere is the collection of points in space which are equidistant from a fixed point. This point is called the center of the sphere. The common distance of the points of the sphere from the center is called the radius.

Spheres are the natural 3-dimensional analog of circles.

The volume of a sphere is $\dfrac{4}{3}\pi r^3$, where r is the radius of the sphere.

The surface area of a sphere is $4\pi r^2$, where r is the radius.

## Fractions of a sphere

In a circle, a sector of measure $\theta$ covers $\frac{\theta}{2\pi}$ the circumference and area of the entire circle. In a sphere, the formula is less obvious. Consider the set of all points on a sphere within angle $\theta$ of a given point (for example, if $\theta = 10^\circ$, then we might have the set of all points on Earth whose latitude is above $80^\circ$ North). The fraction of this encompassed by the entire sphere is

$\text{Fraction} = \frac{1}{2} - \frac{1}{2}\cos\theta$

A special case of this formula is $\theta = 60^\circ$, which tells us that the $30^\circ$ latitude lines of Earth cut the area of their respective hemispheres in half.

From this formula, we can deduce other sphere-related formulas, such as the volume of a cap cut off by a plane.