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Finding an Apothem of a Regular Polygon

Mukulika Mukherjee Feb 15, 2019
Looking for an easy way to calculate the apothem of a polygon? Here's help!

Apothem Fact #1

The apothem of a regular polygon is always the radius of the inscribed circle, i.e., the circle drawn within the polygon.
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.

Calculating the Apothem of a Regular Polygon

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 .

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.

Apothem Fact #2

The apothem of a square is equal to half the length of its side!

Now, let us consider the smaller triangle. It is a right-angled triangle where its height is the apothem of the polygon, its base is equal to half the length of the sides of the polygon, while its hypotenuse is the circumradius of the polygon.
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)
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)

Apothem Fact #3

Since a perpendicular is the shortest distance between two given points, the apothem is the shortest distance between the center of a polygon and any of its sides.

So, we have two formulae for calculating the apothem of a regular polygon. Let's see when to use each of them.
1. To calculate apothem a when the value of the side s of the polygon is given:

Apothem a = s / 2 tan (π/n)


2. To calculate apothem a when the value of the circumradius r of the polygon is given:

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!