Regular polygons

Table of Contents

Understanding Regular Polygons

A polygon is called regular when it is both:

Equilateral – all side lengths are equal.
Equiangular – all interior angles are equal.

So a regular polygon is as “even” as possible: all its sides match, and all its corners look the same.

Common examples of regular polygons include:

Equilateral triangle (3 equal sides, 3 equal angles)
Square (4 equal sides, 4 equal angles)
Regular pentagon (5 equal sides, 5 equal angles)
Regular hexagon (6 equal sides, 6 equal angles)
And so on, for any number $n \ge 3$ of sides

In this chapter we focus on properties that are specific to regular polygons, beyond what applies to general polygons.

Vertex, Side, and Center

A regular polygon with $n$ sides is often called a regular $n$‑gon.

Each corner is a vertex.
Each straight edge between two vertices is a side.
There is a special point inside: the center.

It is the point that is the same distance from every vertex.
In a regular polygon, the center is also the same distance from every side.

That unique “balanced” center is what allows us to define radii, apothems, and use neat formulas.

Interior and Exterior Angles of a Regular Polygon

You will have already seen the general idea of interior angles of polygons. For a regular polygon, all interior angles are equal, so we can find the measure of each one easily.

For an $n$‑sided polygon (an $n$‑gon):

The sum of interior angles is
$$ (n - 2)\times 180^\circ. $$

In a regular $n$‑gon, all $n$ interior angles are equal, so each interior angle is:

$$
\text{Each interior angle} = \frac{(n - 2)\times 180^\circ}{n}.
$$

The exterior angle at a vertex (taking the “outside” angle that forms a straight line with a side) is supplementary to the interior angle, so

$$
\text{Exterior angle} = 180^\circ - \text{Interior angle}.
$$

Using the formula for each interior angle:

\[
\text{Exterior angle}
= 180^\circ - \frac{(n - 2)\times 180^\circ}{n}
= \frac{180^\circ \cdot n - 180^\circ(n-2)}{n}
= \frac{360^\circ}{n}.
\]

So for a regular $n$‑gon:

Each interior angle: $\displaystyle \frac{(n - 2)\times 180^\circ}{n}$.
Each exterior angle: $\displaystyle \frac{360^\circ}{n}$.

A useful consequence: the exterior angles of any convex polygon sum to $360^\circ$, so in a regular polygon:

$n \times \text{(each exterior angle)} = 360^\circ$,
hence $\text{each exterior angle} = 360^\circ / n$.

This also tells you that exactly $n$ copies of the exterior angle “turn” you once around a full circle.

The Center, Radius, and Apothem

A regular polygon fits naturally with a circle.

Draw a circle through all the vertices of a regular polygon:

This is the circumscribed circle.
Its radius is the radius of the polygon, usually denoted $R$.
Every vertex lies on this circle, and the center of the circle is the center of the polygon.

Draw a circle tangent to every side of the polygon:

This is the inscribed circle.
Its radius is called the apothem, usually denoted $a$.
The apothem is the distance from the center to any side, along a perpendicular segment.

So for a regular polygon:

Radius $R$: distance from center to a vertex.
Apothem $a$: distance from center to a side.

Both are the same for all vertices and all sides, respectively, because of the regularity.

Central Angles

At the center of a regular $n$‑gon, connect the center to each vertex. This divides the polygon into $n$ congruent isosceles triangles.

The angle at the center between lines to two neighboring vertices is called a central angle.

The full angle around a point is $360^\circ$.
It is divided evenly into $n$ pieces.

So the central angle for a regular $n$‑gon is:

$$
\text{Central angle} = \frac{360^\circ}{n}.
$$

This gives a useful picture:

Each side of the polygon is a chord of the circumscribed circle.
Each such chord “subtends” (stands opposite) a central angle of $360^\circ / n$.

Side Length, Radius, and Apothem Relations

Because the regular polygon splits into $n$ identical isosceles triangles, you can relate its side length to its radius and apothem using basic trigonometry (sine, cosine, tangent will be treated in trigonometry chapters; here we only quote the relationships).

Let

$n$ be the number of sides,
$s$ be the side length,
$R$ be the radius (center to vertex),
$a$ be the apothem (center to side).

Half of a central triangle is a right triangle with:

Hypotenuse $R$,
One leg $a$,
Other leg $s/2$,
Acute angle at the center of size $\dfrac{180^\circ}{n}$ (half of the central angle).

Using standard trigonometry one gets:

$$ s = 2R \sin\left(\frac{\pi}{n}\right) $$
$$ a = R \cos\left(\frac{\pi}{n}\right) $$
$$ s = 2a \tan\left(\frac{\pi}{n}\right) $$

(Here $\pi$ is used to express angles in radians, where $180^\circ = \pi$ radians.)

These formulas show how the shape of a regular polygon is controlled by a single length (for example, $R$ or $s$).

Perimeter of a Regular Polygon

The perimeter $P$ of a regular $n$‑gon is particularly simple:

There are $n$ equal sides, each of length $s$.

$$
P = n \cdot s.
$$

Combining this with the trigonometric relations from above, you can also express the perimeter in terms of the radius $R$ or the apothem $a$:

Using $s = 2R \sin\left(\frac{\pi}{n}\right)$:
$$ P = n \cdot 2R \sin\left(\frac{\pi}{n}\right). $$
Using $s = 2a \tan\left(\frac{\pi}{n}\right)$:
$$ P = n \cdot 2a \tan\left(\frac{\pi}{n}\right). $$

Area of a Regular Polygon

The area of a general polygon can be found in several ways, but for a regular polygon there is a very convenient formula involving its perimeter and apothem.

Imagine the regular polygon cut into $n$ identical isosceles triangles, each with:

Base $s$ (one side of the polygon),
Height $a$ (the apothem).

The area of one triangle is

$$
\frac{1}{2}\times \text{base}\times \text{height}
= \frac{1}{2} s a.
$$

There are $n$ such triangles, so the total area $A$ is

$$
A = n \cdot \frac{1}{2} s a.
$$

But $n \cdot s$ is the perimeter $P$, so:

$$
A = \frac{1}{2} P a.
$$

This is the main area formula for any regular polygon:

$$ \boxed{A = \dfrac{1}{2} P a} $$

where:

$A$ = area,
$P$ = perimeter,
$a$ = apothem.

If you prefer expressions in terms of $n$ and $s$ only, you can use $P = ns$ and trigonometric relations to obtain

$$
A = \frac{1}{4} n s^2 \cot\left(\frac{\pi}{n}\right),
$$

where $\cot$ is the cotangent function (defined in trigonometry).

Special Regular Polygons

Some small regular polygons are especially common:

Equilateral triangle ($n = 3$):

Each interior angle: $\dfrac{(3-2)\cdot 180^\circ}{3} = 60^\circ$.
Each exterior angle: $\dfrac{360^\circ}{3} = 120^\circ$.

Square ($n = 4$):

Each interior angle: $\dfrac{(4-2)\cdot 180^\circ}{4} = 90^\circ$.
Each exterior angle: $\dfrac{360^\circ}{4} = 90^\circ$.

Regular hexagon ($n = 6$):

Each interior angle: $\dfrac{(6-2)\cdot 180^\circ}{6} = 120^\circ$.
Each exterior angle: $\dfrac{360^\circ}{6} = 60^\circ$.

Because regular hexagons fit together neatly (three interior angles of $120^\circ$ add to $360^\circ$), they are often seen in tilings and natural patterns (like honeycombs).