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Mukulika Mukherjee
Feb 15, 2019

Looking for an easy way to calculate the apothem of a polygon? Here's help!

A polygon is a closed geometrical figure that has three or more sides. A regular polygon is one in which all sides and all interior angles are equal. The apothem of a regular polygon is the line segment that is drawn from its center to the midpoint of any of its sides. The term can also denote the length of the said line segment.

Now, if you're left wondering why we're talking about regular polygons here, then it's only because irregular polygons cannot have an apothem. The length of all apothems of a regular polygon are equal. So, let's calculate the apothem of polygons.

To derive the formulae for the calculation of the apothem of a polygon, let us consider a hexagon.

We know that 180 degrees is equal to **π** radians. Since the angle around a point is equal to 360 degrees, we write it as **2π**.

Since the hexagon has six sides, the angle at the apex of each isosceles triangle within it will be given by 2π/6. For a polygon of**n** sides, it is given by **2π/n**.

Since the hexagon has six sides, the angle at the apex of each isosceles triangle within it will be given by 2π/6. For a polygon of

To find how the apothem "a", the base "b", and the circumradius "r" are related, let us calculate the sin, cosine, and tangent values for the angle **π/n.**

**sin (π/n) = (Base of the triangle / Radius) = (Side of the Polygon/2) / Radius**

Therefore,**Radius = Side / 2 sin (π/n)**

Therefore,

Similarly, we can derive the cosine and tangent values which are given as follows:

**cos (π/n) = (Apothem / Radius)**

Therefore,**Apothem = Radius x cos (π/n)**

**tan (π/n) = (Base / Apothem) = (Side of the polygon/2) / Apothem**

Therefore,**Apothem = Side of the polygon / 2 tan (π/n)**

Therefore,

Therefore,

So, we have two formulae for calculating the apothem of a regular polygon. Let's see when to use each of them.

Apothem a = s / 2 tan (π/n)

Apothem a = r x cos (π/n)

You can use the two given formulae for calculating the apothem of a regular polygon with any number of sides. Have fun!