11.1 – Three Dimensions

Key Terms

• Axis – One of the two lines that form a Cartesian coordinate system.
• The horizontal axis is usually called the x-axis, and the vertical axis is usually called the y-axis.
• The plural of axis is axes.
• Cartesian Coordinate System – A coordinate system formed by two number lines, one horizontal and one vertical.
• They intersect at the zero point of each line.
• The number lines are called axes and are usually labeled the x-axis and y-axis.
• Height – The measurement taken from the bottom to the top of an object.
• Length – A measurement taken horizontally across the longest side of an object.
• Line Segment – A part of a line with endpoints at both ends.
• The symbol AB means “the line segment with endpoints A and B.” It is sometimes called a segment.
• One-Dimensional – Having length but no width or height.
• Perspective – A technique of representing three-dimensional objects and their relationships to each other on a two-dimensional surface.
• Point – The most basic object in geometry, used to mark and represent locations.
• Points have no length, width, or height.
• Solid – An object that has three dimensions: length, width, and height.
• Also called a solid figure or a three-dimensional figure.
• Square – A quadrilateral with four right angles and four congruent sides.
• Squares have all of the properties of parallelograms, rectangles, and rhombi.
• Three-Dimensional – Having length, width, and height.
• Width – The measurement taken from one side of an object to the other side (or front to back).
• Zero-Dimensional – Having no length, width, or height.

Review

Dimensions What Can Be Measured Example Object
Zero Nothing Points
One Length Segments, Lines, Rays
Two Length, Width Polygons, Planes
Three Length, Width, Height Cubes, Solids

Dimensions
• Zero-dimensional: One point defines a point.
• One-dimensional: Two points define a line.
• Two-dimensional: Three non-collinear points define a plane.
• Three-dimensional: Four points define space.

Notes

Item Two Dimensions Three Dimensions
Number of Axes 2 3
Point Notation (x, y) (x, y, z)
Real Examples Maps, Paintings Globes, Sculptures

Three-Dimensional Objects
• Built from 0, 1, and 2 dimensional objects
• Have length, width, and height
• Can be graphed using x, y, and z axes

Examples
• Ex 1. It is true that, in geometry, a solid may exist in three-dimensional space.
• Ex 2. It is true that one can use two-dimensional objects to build three-dimensional objects.
• Ex 3. It is true that many rules concerning two-dimensional geometry have three-dimensional analogues.
• In other words, the rules of two-dimensional geometry can be applied to three-dimensional solids.