Regular Polygon

Regular Polygon

A polygon for which all sides are congruent and all angles are congruent.

Regular Polygon Formulas

n = number of sides
s = length of a side
r = apothem (radius of inscribed circle)
R = radius of circumcircle

Sum of interior angles = (n – 2)·180°

Area = (½)nsr

Three equivalent area formulas for a regular polygon: Area = (1/4)ns²cot(180°/n) = nr²tan(180°/n) = (1/2)nR²sin(360°/n)

A regular pentagon (5-sided polygon) with equal sides and angles, shown as a plain white geometric figure.
Regular Pentagon

A regular hexagon with six equal sides and angles, representing a regular polygon shape.
Regular Hexagon

A regular heptagon (7-sided polygon) with equal sides and angles, no labels visible.
Regular Heptagon

A regular octagon (8-sided polygon) with equal sides and equal interior angles, shown as a geometric example of a regular polygon.
Regular Octagon

A regular polygon with 9 sides (nonagon) showing equal sides and angles, no labels.
Regular Nonagon

Key Formula

\text{Sum of interior angles} = (n - 2) \cdot 180°

\text{Each interior angle} = \frac{(n - 2) \cdot 180°}{n}

\text{Area} = \frac{1}{2} n s r

Where:

$n$ = Number of sides of the regular polygon
$s$ = Length of one side
$r$ = Apothem — the distance from the center to the midpoint of a side (radius of the inscribed circle)
$R$ = Radius of the circumscribed circle (circumcircle) — the distance from the center to a vertex

Worked Example

Problem: Find each interior angle and the area of a regular hexagon with side length 10 cm and apothem approximately 8.66 cm.

Step 1: Identify the number of sides. A hexagon has 6 sides, so n = 6.

n = 6

Step 2: Find the sum of interior angles using the formula.

\text{Sum} = (6 - 2) \cdot 180° = 4 \cdot 180° = 720°

Step 3: Divide by the number of angles to find each interior angle.

\text{Each angle} = \frac{720°}{6} = 120°

Step 4: Calculate the area using A = (1/2)nsr, with s = 10 cm and r = 8.66 cm.

A = \frac{1}{2} \cdot 6 \cdot 10 \cdot 8.66 = \frac{1}{2} \cdot 519.6 = 259.8 \text{ cm}^2

Answer: Each interior angle is 120°, and the area is approximately 259.8 cm².

Another Example

This example works backward from a known interior angle to find the number of sides, which is a common exam question and reverses the direction of the first example.

Problem: A regular polygon has interior angles that each measure 135°. How many sides does it have?

Step 1: Start with the interior angle formula and set it equal to 135°.

\frac{(n - 2) \cdot 180°}{n} = 135°

Step 2: Multiply both sides by n to clear the denominator.

(n - 2) \cdot 180 = 135n

Step 3: Expand the left side and solve for n.

180n - 360 = 135n

Step 4: Subtract 135n from both sides and solve.

45n = 360 \quad \Rightarrow \quad n = 8

Answer: The polygon has 8 sides — it is a regular octagon.

Frequently Asked Questions

What is the difference between a regular and irregular polygon?

A regular polygon has all sides the same length AND all angles the same measure. An irregular polygon fails one or both of these conditions. For example, a rectangle has four equal angles (each 90°) but unless all four sides are also equal, it is irregular — only a square is a regular quadrilateral.

Is a circle a regular polygon?

No, a circle is not a polygon at all because it has no straight sides. However, as the number of sides of a regular polygon increases, its shape gets closer and closer to a circle. This limit relationship is useful in calculus and approximation problems.

How do you find the apothem of a regular polygon?

If you know the side length s and the number of sides n, the apothem is

r = \frac{s}{2\tan(\pi/n)}

. The apothem is the perpendicular distance from the center to the midpoint of any side. It equals the radius of the inscribed circle.

Regular Polygon vs. Irregular Polygon

	Regular Polygon	Irregular Polygon
Definition	All sides congruent and all angles congruent	Sides and/or angles are not all congruent
Interior angle formula	Each angle = (n − 2)·180° / n	Angles vary; no single formula for each angle
Area formula	A = (1/2)nsr using the apothem	Requires other methods (triangulation, coordinates, etc.)
Symmetry	Has n lines of symmetry and rotational symmetry of order n	May have fewer or no lines of symmetry
Examples	Equilateral triangle, square, regular pentagon	Scalene triangle, rectangle (non-square), general quadrilateral

Why It Matters

Regular polygons appear throughout geometry courses — from basic angle problems to tiling patterns and circle approximations. They are essential in real-world design and engineering, such as hexagonal bolts, octagonal stop signs, and tessellations in architecture. Many standardized math tests (SAT, ACT, GCSEs) include questions that require you to calculate interior angles or areas of regular polygons.

Common Mistakes

Mistake: Using (n − 2)·180° as the measure of a single interior angle instead of the sum.

Correction: The formula (n − 2)·180° gives the total sum of all interior angles. To find one angle in a regular polygon, you must divide by n: each angle = (n − 2)·180° / n.

Mistake: Assuming any polygon with equal sides is regular.

Correction: Equal sides alone are not sufficient. A rhombus has four equal sides but its angles are not all equal (unless it is a square), so it is not regular. Both conditions — congruent sides and congruent angles — must hold.

Related Terms

Polygon — General term; regular polygons are a special case
Apothem — Center-to-side distance used in the area formula
Interior Angle — Each angle inside the polygon
Area of a Regular Polygon — Dedicated formula page for computing area
Circumcircle — Circle passing through all vertices of the polygon
Inscribed Circle — Circle tangent to every side; radius equals the apothem
Congruent — Key property — all sides and angles must be congruent
Hexagon — Common regular polygon with 6 sides