Depth

Depth in math is a type of vertical distance. One of the most common everyday examples is the depth of some body of water, which is measured as the perpendicular vertical distance from the surface of the water to some point below the water. In the figure below, the depth between the center of the boat and the bottom of the lake is shown.

Depth definition

Depth is defined as the measure of perpendicular distance downward from a chosen reference point. In the real-life example above, the chosen reference point is the surface of the water and the depth is distance from the surface to the bottom of the lake.

Depth in 3D shapes

In three-dimensional shapes, the definition of depth can sometimes be unclear because it can overlap with other measures such as width and height. Consider the following figures in which the dimensions of length, width, depth, and height are used.

When labeling a rectangle in two dimensions, we typically use the dimensions length and width, where length is the longer side and width is the other side. If we extend this to a rectangular prism in three dimensions, we get the figure above, where depth is the third dimension. Notice in the figure that the depth is indicated as the downward distance measured from the top of the figure; this measure is actually more typically labeled as "height," and the "width" in this figure is more typically labeled as "depth," as shown in the following figure:

In this case, the dimension that was previously labeled as "depth" becomes the "height", where the height is the vertical distance measured upward from the bottom of the figure. They are exactly the same distance, but the way we use the terms and reference the figure changes how we may label the figure. The depth definition in this case then becomes how "deep" into the page the figure goes, rather than describing a vertical distance.

Consider yet another figure, where the orientation of the rectangular prism is changed:

In this figure, we used the dimensions of depth, height, and width (instead of length) to label the rectangular prism. The depth in this case is the same as in the second example: it is the measure of the distance into the page, or how far back into the page the figure goes. In this case, even though height is the longest dimension of the rectangle, hopefully we can agree that it makes more intuitive sense to label that dimension as height rather than length, even though labeling it as length would not actually be incorrect.

From just these few examples, we can quickly see how labeling a 3D figure can get unclear or confusing due to the various ways we can view a figure or choose to label it. Unfortunately, there is no real convention or "correct" way to label a 3D figure. The best we can do is to be clear in our labeling and be explicit in our definitions of the dimensions that we use when referencing a figure.

Length, width, height, depth

From the examples above, we can see that the dimensions of a 3D shape can be labeled using various sometimes overlapping terms. Here we will define these terms as best we can, acknowledging that there is no exact convention and that some may define the terms differently.

• Depth - the distance from front to back of a 3D figure (often interchangeable with width); the perpendicular vertical distance measured downward from a reference point at the top of a figure (often interchangeable with height, where the reference point for height starts at a point at the bottom).


• Length - the longest dimension of a 3D object. If a figure is oriented such that the longest dimension is vertical, the dimension is most commonly labeled as height, rather than length.


• Width - the horizontal distance from side to side of a 3D figure (often interchangeable with depth). Sometimes, it is the horizontal distance from front to back.


• Height - the vertical distance measured upwards from a reference point at the bottom of the figure (often interchangeable with height, where the only difference is the reference point for measurement).

Depth and height

Depth and height are two measures that can be a bit ambiguous, particularly in the labeling of 3D shapes. The International Organization for Standardization (ISO) defines depth and height as follows:

The only real difference between the definitions is the point of reference and the direction of measurement. There are conventions for measurements of distance, but like in the shape examples above, there is some freedom on how to label various distances, as long as they are consistent and clear.