Pentadecagon — Definition, Formula & Examples

A pentadecagon is a polygon with 15 sides and 15 vertices. When all sides and angles are equal, it is called a regular pentadecagon.

A pentadecagon is a 15-gon, a closed plane figure composed of 15 straight line segments (sides) connected end-to-end. In its regular form, each interior angle measures $156°$ and all sides are congruent.

Key Formula

S = (n - 2) \times 180°

Where:

$S$ = Sum of interior angles of the polygon
$n$ = Number of sides (15 for a pentadecagon)

Worked Example

Problem: Find the measure of each interior angle of a regular pentadecagon.

Step 1: Use the interior angle sum formula with n = 15.

S = (15 - 2) \times 180° = 13 \times 180° = 2340°

Step 2: Since a regular pentadecagon has 15 equal interior angles, divide the sum by 15.

\text{Each angle} = \frac{2340°}{15} = 156°

Answer: Each interior angle of a regular pentadecagon measures

156°

Why It Matters

The pentadecagon is historically notable because Euclid showed it can be constructed with a compass and straightedge, one of the few polygons with that property. Understanding polygons with many sides also builds toward the idea that a circle is the limit of a regular polygon as the number of sides increases.

Common Mistakes

Mistake: Confusing a pentadecagon (15 sides) with a pentagon (5 sides) because both names start with "penta."

Correction: "Penta" means 5 and "deca" means 10. A pentadecagon literally means a 5-and-10-gon, so it has 15 sides. A pentagon has only 5.