## Spheres

Spheres are three-dimensional closed surfaces.

A sphere is a set of points in three-dimensional space equidistant from a point called the center. The radius of the sphere is the distance from the center to the points on the sphere. Spheres are not polyhedra. Of all shapes, a sphere has the smallest surface area for its volume.

The volume of a sphere is four-thirds times pi times the radius cubed.
$$V=\frac { 4 }{ 3 } \pi { r }^{ 3 }$$
(Volume of a sphere: r = radius)

Note: A cross section of a geometric solid is the intersection of a plane and the solid.

The surface area of a sphere is four times the area of the largest cross-sectional circle (called the great circle). SA = 4πr2 + πd2

A great circle is the largest circle that can be drawn on a sphere. Such a circle will be found when the cross-sectional plane passes through the center of the sphere.

The equator is an examples of a great circle. Meridians (passing through the North and South poles) are also great circles.
The shortest distance between two points on a sphere is along the arc of the great circle joining the points. The shortest distance between points on any surface is called a geodesic. In a plane, a straight line is a geodesic. On a sphere, a great circle is a geodesic.

What happens when planes intersect with spheres?
1. The intersection of a plane and a sphere is a circle. 2. If two planes are equidistant from the center of a sphere (and intersecting the sphere), the intersected circles are congruent. A hemisphere is the half sphere formed by a plane intersecting the center of a sphere. 