The word dimension is related to the word measure. It is used in two ways in geometry.

• It is used to specify a measurement:

“What are the dimensions of this rectangle?” or “Build a rectangular prism that has these dimensions.”

• It is also used to count the (mutually perpendicular) directions that an object can be measured.

“A rectangle has two dimensions” or “This is a three-dimensional figure” or “How many dimensions does a point have?”
“Length is a one-dimensional measure, but area is a two-dimensional measure.”

Counting mutually perpendicular directions an object can be measured

• A point has zero dimensions: there is nothing to measure; a point just specifies a location, but has no size.
• A line segment has one dimension: we can measure its length, but it has no width or thickness or any other measurable feature.
• A rectangle has two dimensions: we can measure its length and, perpendicular to that, its width. The interior of a triangle or oval is also two-dimensional. Though we don’t think of these as having “length” or “height,” they cover a region that has extent in not just one direction but two.

We can measure a circle in any direction. Why do we count it as only two-dimensional? Because, once a measurement is made in one direction, there is only one direction perpendicular to that for another measurement to be made — two directions, in total.

• A rectangular prism has three dimensions: we can measure its length; perpendicular to that, we can measure its width; and perpendicular to both of those dimensions, we can make a third length-measurement, the prism’s height or depth.

Specifying dimensions

Dimension can refer to the actual one-dimensional measurements (measurements that can be made with a ruler) one makes on a figure: e.g., “The dimensions of this rectangle are 3 inches by 4 inches.”

As always, dimension should be used only to refer to measurements that are made at right angles to each other (like the length, width, and height of a rectangular prism). Generally, the word dimension is used to designate actual measurements only when one is referring to rectangles or rectangular prisms.

• A triangle’s side lengths may be 6, 8, and 10 inches, but it would be non-standard to refer to the dimensions as 6, 8, and 10 inches.
• Used in this way, dimension refers only to the kind of measurements that can be made with a ruler (that is, lengths). So although area is a measurement that can be made on a rectangle, it would be non-standard in geometry to refer to it as a “dimension” of the rectangle.

We live in a three-dimensional world

Everything we see — a piece of paper, even a single dot of ink or pencil on a piece of paper — is three-dimensional. The paper has not only length and width, but thickness (in a direction perpendicular to both the length and the width). Even the ink has thickness as well as length and width.

What about the rectangles we draw? Aren’t they two-dimensional? Well, the rectangles in our mind are two dimensional. We use drawings to represent those mental pictures and help us think and communicate about them. But drawings are just drawings. If we looked under a powerful microscope, the “straight” pencil line would be rough and jagged and full of holes. Drawings are imperfect in many ways, but good enough to do their purpose, which is to represent our ideas. Only the ideas can be mathematically perfect.

It is fine, in teaching, to use a piece of paper to suggest the idea of a plane, or to “cut out a triangle,” even though, in both cases, the paper has thickness and is therefore three-dimensional (and so not really a plane or a triangle at all). But, for children who have begun to distinguish and think about two-dimensional vs. three-dimensional objects, it may also be worth saying “Of course, this piece of paper has thickness, so really it is three-dimensional, too, like everything we can see.” If children challenge the idea that everything we can see is three-dimensional, that is a lovely opportunity for a discussion!

What’s in a word?

The word dimension comes from Latin di- (intensive) + -mens measure.

Mens is one of several forms derived from the Indo-European root, me-, meaning ‘measure,’ which also gives us Meter and Measure. Related non-mathematical words include commensurate, immense (unmeasurable). The sense of time measure gives rise to moon and month (a ‘moonth’ of a year!), and monthly events (menstruation and related words), and semester (measured in two parts), trimester (measured in three parts), etc.

Incommensurable

Incommensurable quantities cannot be measured with repeated uses of the same unit. A rod of length and a rod of length can both be “measured” by a rod of length , because , copied a whole number of times fits each of those lengths exactly: it fits twice into and three times into . Similarly, 4.139001 and 6.0003 can both be “measured” by 0.000001, which fits 4139001 times in 4.139001 and fits 6000300 times into 6.0003. Any two rational numbers (numbers that can be expressed as one whole number divided by another) are commensurable.

It takes a little bit of algebra or geometry to prove that √2 and 1 are not commensurable — that is, that there is no length, no matter how tiny, that will fit a whole number of times into both √2 and 1.