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Measurement: Length, width, height, depth

Length, width, height, depth


People often ask about the "correct" use of length, width, height, and depth.

In the most common contexts in elementary school, mathematics does not have a formal definition for these terms, nor does it have rules about "proper" use. But natural usage does follow some conventions. (See below.)
In some contexts, two of these terms -- length and height -- do also have a specific mathematical meaning. Length, when referring to measurement in one dimension (as in the length of a line segment or a or a piece of string), has a specialized meaning. So does height when it is used in conjunction with base. (See those articles.)

Correct usage

One dimension

Mathematics uses theterm length to name the measure of one-dimensional objects like theseImage:length.png. Though we might straighten out a piece of string to measure its length against a ruler, the length of the string is the same even when it is curled or folded. (For more about length and distance see either of those topics.)

Two dimensions

We call the measure of a two-dimensional object its area. We may also measure the length of the boundary of the two-dimensional object -- the line segments or curves that enclose it. The length of the entire boundary is called the perimeter or circumference.

Question: Rectangles may be drawn in various sizes, shapes, and positions. Do we label the two dimensions length and width; or do we use width and height; or even length and height? Is there a best way?

Answer: There is more than one correct way! Any pair of these words can be used, as long as your words are used sensibly.

Length: If you use the word length, it should certainly be for the longest sides of the rectangle. Think of how you would describe the distance along a road: it is the long distance, the length of the road. (The words along, long, and length are all related.) The distance across the road tells how wide the road is from one side to the other. That is the width of the road. (The words wide and width are related, too.)

When a rectangle is drawn "slanted" on the page, like this Image:SlantedRectangle.png, it is usually clearest to label the long side "length" and the other side "width," as if you were labeling a road.

Height: If the rectangle is drawn with horizontal and vertical sides, people often use the word height to describe how high (how tall) the rectangle is. It is then perfectly correct to describe how wide the rectangle is from side to side by using the word width.

As you can see, when height and width are used together, either one can be greater. (When the word height is used with base, it has a different meaning.)

Three dimensions

We call the measure of athree-dimensional object its volume. Because edges are one-dimensional, and faces are two-dimensional, their measures are length and area, respectively.

As for naming the dimensions of a three-dimensional figure, the only rule is be sensible and clear.

When the figure is "level," height (if you use the word) refers consistently to how tall the figure is, even if that dimension is greatest; length (if you use the word) refers to the longer of the other two dimensions. But you may also refer to the other dimensions as width and depth (and these are pretty much interchangeable, depending on what "seems" wide or deep about the figure).

When the figure is not "level" Image:CuisenaireRodNonLevel.png people cannot know what is meant by width, depth, or height without being told (or without labels), although length is generally still assumed to mean the direction in which the figure is "longest."

About the words

Length, width, height, and depth are nouns are derived from the adjectives long, wide, high, and deep, and follow a common English pattern that involves a vowel change (often to a shorter vowel) and the addition of th. (The lone t in height is modern. Obsolete forms include heighth and highth, and it is still common to hear people pronounce it that way.)

wide     deep     high     long     broad
width     depth     height     length     breadth

Other English adjective-noun pairs are related in this way, too: e.g., hale as in "hale and hearty" and health (but hale, except in that expression, has now been replaced almost totally with "healthy").