Calculator Inputs
Use the responsive input grid below. It shows three columns on large screens, two on smaller screens, and one on mobile.
Example Data Table
| Example | Sides | Input Mode | Main Input | Root Vertex | Triangles | Distinct Triangulations |
|---|---|---|---|---|---|---|
| Regular hexagon | 6 | Known side length | 10 units | V1 | 4 | 14 |
| Regular octagon | 8 | Known circumradius | 12 units | V3 | 6 | 132 |
| Unit-circle decagon | 10 | Unit circle model | Radius = 1 | V2 | 8 | 1,430 |
Formula Used
1) Distinct triangulations of a convex n-gon
Triangulations = Catalan(n - 2) = (1 / (n - 1)) × C(2n - 4, n - 2)
This gives the number of different ways to split a convex polygon into non-overlapping triangles using diagonals that do not cross.
2) Triangles and diagonals in one fan triangulation
Triangles = n - 2
Diagonals in one triangulation = n - 3
A fan triangulation fixes one vertex and connects it to all non-adjacent vertices.
3) Total diagonals in the polygon
Total diagonals = n(n - 3) / 2
This counts every internal diagonal of the full polygon, not only the selected fan.
4) Regular polygon measurements
Perimeter = n × side length
Area from side length = n × s² / (4 × tan(π / n))
Circumradius from side length = s / (2 × sin(π / n))
Area from circumradius = n × R² × sin(2π / n) / 2
How to Use This Calculator
- Enter the number of polygon sides. Use values from 3 to 100.
- Choose your dimension mode: known side length, known circumradius, or a unit-circle model.
- Provide the main measurement when needed, then choose the root vertex for the fan triangulation.
- Set a decimal precision and optional unit label such as cm, m, or units.
- Click Generate triangulation to show the result panel above the form.
- Review the graph, summary table, triangle table, and coordinate table.
- Use the CSV button for spreadsheet-ready data and the PDF button for a shareable report.
FAQs
1) What does this triangulation generator calculate?
It calculates the number of triangles in a fan split, diagonals in that split, total polygon diagonals, distinct triangulations, regular polygon measurements, and vertex coordinates for the graph.
2) Does it work for every polygon?
The counting formulas apply to convex polygons. The drawing and coordinate output use a regular convex polygon model, so irregular or self-intersecting shapes are not represented here.
3) Why is the triangulation count so large?
Triangulation counts follow Catalan numbers, which grow very fast. Even moderate side counts produce huge exact values, so the calculator also shows a scientific-notation style approximation.
4) What is a fan triangulation?
A fan triangulation starts from one chosen root vertex and connects it to all non-adjacent vertices. That creates a valid triangulation with exactly n minus two triangles.
5) Why are there two diagonal counts?
One count shows diagonals used by the selected fan triangulation. The other count shows every possible diagonal inside the polygon, including diagonals not used in the chosen fan.
6) Can I export the results?
Yes. The page includes CSV export for data analysis and PDF export for reports, homework handouts, audits, or documentation snapshots.
7) Why do the triangle areas change?
In a regular polygon, a fan from one vertex usually creates triangles with different shapes. Their individual areas may vary, but their total equals the regular polygon area.
8) Can I use side length or radius?
Yes. You can input a side length, a circumradius, or use a unit-circle model. The calculator converts values and then builds the same regular polygon framework.