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=43πr3
(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.