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Triacontagon — Definition, Formula & Examples

A triacontagon is a polygon with 30 sides and 30 vertices. Like all polygons, it can be regular (all sides and angles equal) or irregular.

A triacontagon is a closed, planar figure composed of 30 straight line segments (sides) meeting at 30 vertices. A regular triacontagon has all sides congruent and all interior angles congruent, with each interior angle measuring 168°168°.

Key Formula

S=(n−2)×180°S = (n - 2) \times 180°
Where:
  • SS = Sum of all interior angles
  • nn = Number of sides (30 for a triacontagon)

Worked Example

Problem: Find the measure of each interior angle of a regular triacontagon.
Step 1: Use the interior angle sum formula with n = 30.
S=(30−2)×180°=28×180°=5040°S = (30 - 2) \times 180° = 28 \times 180° = 5040°
Step 2: Since the triacontagon is regular, divide the total by 30 to get each angle.
Each angle=5040°30=168°\text{Each angle} = \frac{5040°}{30} = 168°
Answer: Each interior angle of a regular triacontagon measures 168°168°.

Why It Matters

Polygons with many sides like the triacontagon approximate circles closely, which matters in engineering and computer graphics when curved shapes must be rendered using straight segments. Studying high-sided polygons also reinforces the general interior angle sum formula, a staple of geometry courses.

Common Mistakes

Mistake: Confusing the number of diagonals with the number of sides.
Correction: A 30-sided polygon has 30(30−3)2=405\frac{30(30-3)}{2} = 405 diagonals — far more than its 30 sides. Use the diagonal formula n(n−3)2\frac{n(n-3)}{2} separately.