Pentagram — Definition, Formula & Examples
A pentagram is a five-pointed star shape formed by drawing the five diagonals of a regular pentagon. It creates a smaller regular pentagon in its center and five triangular points around the outside.
A pentagram is a non-convex polygon constructed by connecting each vertex of a regular pentagon to the two non-adjacent vertices, producing a star polygon with Schläfli symbol {5/2}. The interior angles at the five points each measure 36°.
Key Formula
Where:
- = The angle at each of the five star points
- = The total sum of all five point angles
How It Works
To draw a pentagram, start with a regular pentagon and draw a straight line from each vertex to the vertex two positions away. You will draw exactly five lines, and they will intersect inside the pentagon to form a smaller regular pentagon. The five outer triangular points are golden gnomons — isosceles triangles whose sides relate to each other by the golden ratio, approximately 1.618.
Worked Example
Problem: A regular pentagram has an outer edge length (side of the original pentagon) of 10 cm. Find the length of one full diagonal line of the star.
Step 1: In a regular pentagon, each diagonal relates to the side length by the golden ratio φ ≈ 1.618.
Step 2: Multiply to find the diagonal length.
Answer: Each diagonal line of the pentagram is approximately 16.18 cm long.
Why It Matters
The pentagram appears in art, architecture, and nature (such as the cross-section of an apple). It is one of the most accessible examples of the golden ratio in geometry, making it a useful bridge between polygon properties and the study of irrational numbers.
Common Mistakes
Mistake: Confusing a pentagram with a regular pentagon.
Correction: A pentagon is a five-sided polygon. A pentagram is the star figure formed by extending or connecting the diagonals of that pentagon. They are related but distinct shapes.