# Distance

Pronunciation: /ˈdɪs.təns/ Explain

Distance is a measure in one dimension. By convention distance is always positive. If one takes a line and marks it off in equally spaced units, one can measure one-dimensionally. A directed distance is a distance that can be positive or negative.

When describing multi-dimensional objects, such as rectangles (2-dimensional) or solids (3-dimensional), length is the measure of the longest dimension.

Width or breadth measure at right angles to the length. Height measures vertically at right angles to both length and width.

AbbreviationUnit of Measure for DistanceEquals
-micrometer0.000001 m = 10-6 m
mmmillimeter0.001 m = 10-3 m
cmcentimeter0.01 m = 10-2 m
-decimeter0.1 m = 10-1 m
mmeter1 m
-decameter10 m = 101 m
-hectometer100 m = 102 m
kmkilometer1000 m = 103 m
-megameter1000000 m = 106 m
gmgigameter1000000000 m = 109 m
English System
in or "inch1/12 ft ≈ 0.0254m
ft or 'foot1 ft ≈ 0.3048m
ydyard3 ft ≈ 0.9144m
mimile5280 ft ≈ 1609.344m
Table 1: Units of Measure.

### Equality of Distance

 Click on the blue points and drag them to change the figure. Points B and C are equidistant from A. Manipulative 1 - Equidistant Created with GeoGebra.

Two points that are the same distance from a reference point are said to be equidistant from each other. When distance is taken to be positive, the distance from point A to point B is the same as the distance from point B to point A.

 Click on the blue points and drag them to change the figure. The distance from A to B is the same as the distance from B to A. Manipulative 2 - Equidistant Created with GeoGebra.

### Calculation of Distance

Point to Point

 Click on the blue points and drag them to change the figure. To calculate the distance between two points, use the distance formula: C2 = A2 + B2. Manipulative 3 - Calculating the Distance Between Two Points Created with GeoGebra.

The calculation of the distance from one point to another point uses the distance formula where and are the coordinates of the two points.

Point to Line

 Click on the blue points and drag them to change the figure. Manipulative 4 - Calculating the Distance Between a Point and a Line. Created with GeoGebra.

The distance from point C to a line AB is defined as the shortest distance from the point to any point on the line. The shortest distance is found by constructing a new line, say CD that passes through point C perpendicular to line AB. So the first thing to do when calculating the distance from a point to a line is find the equation of this line.

The slope of the new line CD is equal to the reciprocal of the slope of line AB. You can use the point-slope form of a line to find the equation of the line CD.

Once the equation of line CD is found, find the coordinates of the intersection of line AB and CD, which is point D. Since the coordinates of D are the solution of the linear system containing line and , solve the linear system.

Once the coordinates of D have been calculated, all that is left to do is find the distance from C to D. Use the algorithm for the distance between two points.

Line to Line

 Click on the blue points and drag them to change the figure. Manipulative 5 - Calculating the distance between two lines. Created with GeoGebra.

The distance between lines make sense only for lines that are always the same distance apart. This means that the distance between lines only makes sense for parallel lines.

The distance from line AB to a line CD is defined as the shortest distance from a point on CD to any point on the line AB. After selecting an arbitrary point on CD, follow the algorithm for finding the distance between a point and a line.

1. McAdams, David E.. All Math Words Dictionary, distance. 2nd Classroom edition 20150108-4799968. pg 63-64. Life is a Story Problem LLC. January 8, 2015. Buy the book

• McAdams, David E.. Directed Distance. allmathwords.org. Life is a Story Problem LLC. 3/12/2009. https://www.allmathwords.org/en/d/directeddistance.html.

McAdams, David E. Distance. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/d/distance.html.

### Revision History

4/20/2019: Updated equations and expressions to the new format (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)