How to find the interior and exterior angles of a regular 13-sided polygon
Polygons come in all shapes and sizes, but from simple triangles to complex 13-sided figures known as tridecagons or triskaidecagons - calculating their interior and exterior angles is straightforward when you understand the underlying formulas.
In this guide, we’ll explain:
✅ How to find the interior angles of a regular 13-sided polygon
✅ How to calculate the exterior angles
✅ Related polygon properties, such as sum of angles and real-world applications
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Understanding a 13-sided polygon (Tridecagon)
A regular 13-sided polygon (tridecagon) has:
- 13 equal sides
- 13 equal angles
- Symmetry
Unlike irregular polygons, where angles and sides vary in length, a regular tridecagon allows us to use standard geometric formulas for calculations.
How to find the interior angles of a regular 13-sided polygon
Step 1: Use the sum of interior angles formula
The sum of the interior angles of any polygon is found using the formula:
Where n is the number of sides.
For a 13-sided polygon:
So, the total sum of all interior angles in a tridecagon is 1980°.
Step 2: Find each interior angle (for a regular 13-sided polygon)
Since a regular tridecagon has equal angles, we divide the total sum by 13:
Each interior angle of a regular 13-sided polygon is ≈ 152.31° (rounded to two decimal places).
How to find the exterior angles of a regular 13-sided polygon
Step 1: Use the exterior angle formula
The exterior angle of any regular polygon is given by:
For a 13-sided polygon:
So, each exterior angle of a regular tridecagon is ≈ 27.69°.
Step 2: Check your answer using the interior-exterior angle rule
For any polygon:
Checking our values:
✔️ The calculation is correct!
Summary of key polygon formulas
| Property | Formula | 13-sided polygon calculation |
|---|---|---|
| Sum of interior angles | (n−2)×180∘(n - 2) \times 180^\circ | (13−2)×180=1980∘(13 - 2) \times 180 = 1980^\circ |
| Each interior angle (Regular Polygon) | Sum of interior anglesn\frac{\text{Sum of interior angles}}{n} | 1980∘13≈152.31∘\frac{1980^\circ}{13} \approx 152.31^\circ |
| Each exterior angle (Regular Polygon) | 360∘n\frac{360^\circ}{n} | 360∘13≈27.69∘\frac{360^\circ}{13} \approx 27.69^\circ |
Other properties of a 13-sided polygon
1. Number of diagonals in a polygon
The number of diagonals in any polygon is given by:
For a 13-sided polygon:
A regular tridecagon has 65 diagonals.
2. Sum of all polygon's exterior angles
The sum of exterior angles for any polygon is always 360°, no matter the number of sides!
3. Real-world applications of 13-sided polygons
While 13-sided polygons aren’t commonly seen in architecture or engineering, they appear in:
- Coin designs (Some countries use heptagonal and other multi-sided coins for better grip and anti-counterfeiting)
- Geometry problems in advanced mathematics and tessellations
- Symmetric designs in tiling and artistic patterns
A simple formula for calculating angles
A 13-sided polygon, or tridecagon, may sound complex, but calculating its angles is easy with the right formulas. Remember:
- Each interior angle in a regular tridecagon is ≈ 152.31°
- Each exterior angle is ≈ 27.69°
- The sum of interior angles is 1980°
- The sum of exterior angles is always 360°