# Apothem

The apothem is the smallest distance that can be noticed between the center of the figure and any of its sides, being represented through a segment.

In the case of a regular polygon (one that has all its sides and interior angles of the same measure), the apothem has as its extremes the center of the figure and the midpoint of any of its sides.

That is, in the regular polygon, the intersection between the apothem and the side of the geometric figure determines the division of the side into two equal parts.

Likewise, the apothem and the side of the regular polygon are perpendicular, that is, when they intersect they form four right angles or 90º.

As we can see in the figure below, in addition, the apothem (which is the segment FG) is the center of the circumscribed circumference of the polygon, that is, that it contains it.

For example, in the image above, the apothem is the FG segment, while the GI segment is known as the sagite.

An additional fact to take into account is that the apothem in a three-dimensional figure such as the pyramid is the segment that joins the vertex with the midpoint of any of the sides that make up the base of the polyhedron.

## Apothem formula

The apothem formula can be calculated, in the case of a regular polygon, taking the Pythagorean theorem as a reference.

Let's look again at the figure above, the segment FG is the apothem, and the segment AG is half the side of the polygon. Likewise, the segment FA is the radius of the circumference circumscribed to the figure.

So, we have a right triangle where the hypotenuse is the radius of the circumscribed circle (r), while the legs are the apothem (a) and the segment AG that measures half the side (L / 2).

Then, remembering the Pythagorean theorem, the hypotenuse squared is equal to the sum of each of the legs squared. Then we clear the apothem.

It is worth mentioning that this formula is to calculate the apothem of a regular polygon.

## Example of apothem

Suppose we have a polygon that is inscribed in a circle with a radius that measures 17 meters. Also, the side of the figure is 20 meters. What is the length of the apothem of the figure?

The apothem of this polygon is 13.7477 meters.