Star Polygon — Definition, Formula & Examples

A star polygon is a polygon formed by connecting evenly spaced points on a circle in a pattern that skips over one or more points each time, so the sides cross each other and create a star shape.

A star polygon, denoted $\{n/k\}$ , is a figure formed by connecting every $k$ -th point of $n$ equally spaced points on a circle, where $n$ and $k$ are positive integers with $k \geq 2$ and $\gcd(n, k) = 1$ , producing a single closed path whose edges intersect.

Key Formula

\{n/k\}

Where:

$n$ = Number of equally spaced points on the circle (must be at least 5 for a star)
$k$ = Step size — how many points you skip plus one (must satisfy 2 ≤ k < n/2 and gcd(n, k) = 1)

How It Works

To draw a star polygon, place

n

equally spaced points around a circle and number them 0 through

n - 1

. Starting at point 0, draw a straight line to point

k

, then from point

k

to point

2k

(mod

n

), and continue until you return to point 0. For example, a classic five-pointed star is

\{5/2\}

: you connect every 2nd point out of 5. The value of

k

must be greater than 1 (otherwise you just get a regular convex polygon) and

\gcd(n, k)

must equal 1 so the path visits every point before closing.

Worked Example

Problem: Construct a {5/2} star polygon and find the measure of each point angle (the angle at each tip of the star).

Step 1: Place 5 equally spaced points on a circle and label them 0, 1, 2, 3, 4.

Step 2: Connect every 2nd point: 0→2→4→1→3→0. This traces a five-pointed star (a pentagram).

Step 3: The point angle at each tip of a {n/k} star polygon is given by the formula below. Substitute n = 5 and k = 2.

\text{Point angle} = \frac{180(n - 2k)}{n} = \frac{180(5 - 4)}{5} = \frac{180}{5} = 36°

Answer: Each tip of a {5/2} star polygon measures 36°.

Why It Matters

Star polygons appear in flags, religious symbols, and architectural designs worldwide. Understanding them deepens your knowledge of polygon properties like concavity and symmetry, topics that come up throughout geometry courses.

Common Mistakes

Mistake: Confusing a star polygon with any star-shaped drawing. Students sometimes think any concave shape with points is a star polygon.

Correction: A true star polygon requires equally spaced vertices on a circle connected by a consistent step size

k

, forming a single closed path with self-intersecting sides.