# Difference between revisions of "Sphere"

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In a sphere, the formula is less obvious. Consider the set of all points on a sphere within angle <math>\theta</math> of a given point (for example, if <math>\theta = 10^\circ</math>, then we might have the set of all points on Earth whose latitude is above <math>80^\circ</math> North). The fraction of this encompassed by the entire sphere is | In a sphere, the formula is less obvious. Consider the set of all points on a sphere within angle <math>\theta</math> of a given point (for example, if <math>\theta = 10^\circ</math>, then we might have the set of all points on Earth whose latitude is above <math>80^\circ</math> North). The fraction of this encompassed by the entire sphere is | ||

− | <math>Fraction = \frac{1}{2} - \frac{1}{2}\cos\theta</math> | + | <math>\text{Fraction} = \frac{1}{2} - \frac{1}{2}\cos\theta</math> |

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

## Latest revision as of 18:21, 22 February 2019

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 , where r is the radius of the sphere.

The surface area of a sphere is , where r is the radius.

## Fractions of a sphere

In a circle, a sector of measure covers 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 of a given point (for example, if , then we might have the set of all points on Earth whose latitude is above North). The fraction of this encompassed by the entire sphere is

A special case of this formula is , which tells us that the 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.