Polygon Diagonals Calculator

What Is a Polygon Diagonal?

A diagonal of a polygon is a line segment that connects two non-adjacent vertices. In other words, it is a line drawn inside the polygon from one corner to another that is not a side of the polygon. A triangle (3 sides) has no diagonals, while a quadrilateral (4 sides) has exactly 2.

How to Calculate the Number of Diagonals

The number of diagonals in a polygon with n sides is given by the formula:

Why does this formula work?

• Each vertex can connect to \(n - 3\) other vertices via diagonals (excluding itself and its 2 adjacent vertices).
• There are \(n\) vertices, giving \(n(n-3)\) connections.
• Each diagonal is counted twice (once from each end), so we divide by 2.

Sum of Interior Angles

The sum of all interior angles of an n-sided polygon is:

This comes from the fact that any polygon can be divided into \(n - 2\) triangles by drawing diagonals from one vertex, and each triangle has interior angles summing to 180°.

Each Interior Angle of a Regular Polygon

If all sides and angles are equal (a regular polygon), each interior angle is:

Common Polygon Diagonals Reference

How to Use This Calculator

1. Enter the number of sides (n) of your polygon, from 3 to 1,000.
2. Click Calculate Diagonals to see the results.
3. View the interactive polygon diagram showing vertices, sides, and all diagonals.
4. Read the step-by-step calculation breakdown with formulas.
5. Explore the properties table for detailed polygon measurements.

Real-World Applications

• Architecture: Understanding diagonal bracing in polygon-shaped structures for structural integrity.
• Computer Graphics: Polygon triangulation (dividing polygons into triangles) relies on diagonals.
• Network Design: Calculating the number of direct connections in a polygon-topology network.
• Game Development: Collision detection and mesh generation use polygon diagonal decomposition.
• Mathematics Education: Teaching combinatorics and geometric reasoning.

Frequently Asked Questions

What is the formula for the number of diagonals in a polygon?

The formula is \(D = \frac{n(n-3)}{2}\), where \(n\) is the number of sides. This works because each vertex connects to \(n-3\) non-adjacent vertices, and dividing by 2 avoids double-counting.

How many diagonals does a hexagon have?

A hexagon (6 sides) has \(\frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9\) diagonals.

Does a triangle have any diagonals?

No. A triangle has 0 diagonals because every vertex is adjacent to every other vertex — there are no non-adjacent vertex pairs to connect.

What is the sum of interior angles of a polygon?

The sum is \((n-2) \times 180°\). For example, a pentagon (5 sides) has \((5-2) \times 180° = 540°\).

What polygon has the most diagonals?

There is no maximum — the number of diagonals grows quadratically. A 100-gon has 4,850 diagonals, while a 1,000-gon has 498,500 diagonals.