## 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 = 4πr ^{2} + πd^{2}**

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.