Before we can discuss volume, we have to know some basic facts about 3- dimensional space. The first order of business is to set up a coordinate system in 3-space. To this end, we want three lines passing through a point O so that they are mutually perpendicular.

Wu, PCC

Just as in the case of 2 dimensions, “congruence” in 3 dimensions is based on isometries, or distance-preserving maps. Every isometry is a combination of the three basic isometries.

### Basic Isometries in 3 Dimensions

1. Rotation of degrees around a line.
2. Translation by .
3. Reflection across a plane.

### Distance formula in 3D

The distance between two points and is given by ### Cavalieri’s principle

If two solids are placed between two parallel planes and if the areas of the two planar regions cut out in the solids by any plane parallel to the top and bottom planes are always equal, then the volumes of the two solids are also equal.

## Cylinders Right cylinder of height h over a region R, Cylinder of height h over a region R.

The volume of a cylinder is given by , where is the area of the base and is the height (distance between top and bottom

## Cones

If is a region of a plane and is a point not on the plane containing , then the union of all the segments joining the point to a point in the base is called a cone with base and vertex . The height h of the cone is the distance from to the plane containing .

• If is a circle then the cone is called a circular cone. If, in addition, the vertex is located on the line perpendicular to the circular base at its center, then we call is a right circular cone.
• If is a square we call the cone a pyramid.

The volume of a cone with base and height is .

The volume of a sphere with radius is 