Apothemof a Regular Polygon
Definition: A line segment from the center of a regular polygon to the midpoint of a side.
Try this Adjust the polygon below by dragging any orange dot, or alter the number of sides. Note the behavior of the apothem line shown in blue.

The apothem is also the radius of the incircle of the polygon. For a polygon of n sides, there are n possible apothems, all the same length of course. The word apothem can refer to the line itself, or the length of that line. So you can correctly say 'draw the apothem' and 'the apothem is 4cm'.

Each formula below shows how to find the length of the apothem of a regular polygon. Use the formula that uses the facts you are given to start.

Apothem given the length of a side.

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the apothem length is given by the formula:

where
s  is the length of any side
n  is the number of sides
tan  is the tangent function calculated in degrees (see Trigonometry Overview)

Apothem given the radius (circumradius)

If you know the radius (distance from the center to a vertex):

where
r  is the radius (circumradius) of the polygon
n  is the number of sides
cos  is the cosine function calculated in degrees (see Trigonometry Overview)

Irregular Polygons

Since irregular polygons have no center, they have no apothem. In the figure above, uncheck the "regular" checkbox and note how there can be no center or apothem.

Related polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons