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Undecagon

Undecagon

A polygon with eleven sides.

 

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Undecagon

A regular undecagon (11-sided polygon) with equal sides and equal angles.

Regular Undecagon

Key Formula

S=(n−2)×180∘=(11−2)×180∘=1620∘S = (n - 2) \times 180^\circ = (11 - 2) \times 180^\circ = 1620^\circ
Where:
  • SS = Sum of the interior angles of the undecagon
  • nn = Number of sides, which is 11 for an undecagon

Worked Example

Problem: Find each interior angle and each exterior angle of a regular undecagon.
Step 1: Use the interior angle sum formula for any polygon with n sides.
S=(n−2)×180∘S = (n - 2) \times 180^\circ
Step 2: Substitute n = 11 to find the total sum of interior angles.
S=(11−2)×180∘=9×180∘=1620∘S = (11 - 2) \times 180^\circ = 9 \times 180^\circ = 1620^\circ
Step 3: Since a regular undecagon has all equal angles, divide the sum by 11 to find one interior angle.
Each interior angle=1620∘11≈147.27∘\text{Each interior angle} = \frac{1620^\circ}{11} \approx 147.27^\circ
Step 4: Each exterior angle is the supplement of the interior angle. Alternatively, the exterior angles of any convex polygon sum to 360°.
Each exterior angle=360∘11≈32.73∘\text{Each exterior angle} = \frac{360^\circ}{11} \approx 32.73^\circ
Answer: Each interior angle of a regular undecagon is approximately 147.27°, and each exterior angle is approximately 32.73°.

Another Example

Problem: A regular undecagon has a side length of 5 cm. Find its perimeter and the number of diagonals.
Step 1: The perimeter of a regular polygon is the number of sides times the side length.
P=11×5=55 cmP = 11 \times 5 = 55 \text{ cm}
Step 2: Use the diagonal formula for a polygon with n sides.
D=n(n−3)2=11(11−3)2=11×82=44D = \frac{n(n - 3)}{2} = \frac{11(11 - 3)}{2} = \frac{11 \times 8}{2} = 44
Answer: The perimeter is 55 cm, and the undecagon has 44 diagonals.

Frequently Asked Questions

What is another name for an 11-sided polygon?
An 11-sided polygon is most commonly called an undecagon (from Latin) or a hendecagon (from Greek). Both names refer to the same shape. 'Hendecagon' is sometimes preferred in formal mathematical writing because it uses Greek roots, which is more consistent with the naming of other polygons.
Can you construct a regular undecagon with a compass and straightedge?
No. A regular undecagon cannot be constructed using only a compass and straightedge. This is because 11 is not a Fermat prime of the form required by the Gauss–Wantzel theorem for constructible regular polygons. You would need a neusis construction or other specialized tools to draw one precisely.

Undecagon (11 sides) vs. Dodecagon (12 sides)

An undecagon has 11 sides with an interior angle sum of 1620°, while a dodecagon has 12 sides with an interior angle sum of 1800°. Each interior angle of a regular dodecagon is exactly 150°, a clean number, whereas the regular undecagon's interior angle (≈ 147.27°) is not a whole number. Dodecagons appear more frequently in everyday life — for example, some coins and clock faces — while undecagons are relatively rare in practical design.

Why It Matters

The undecagon is a useful example when studying polygon properties because its 11 sides produce non-integer angle measures, reinforcing that angle formulas work for any value of n, not just convenient ones. It also appears in discussions of constructibility, since 11 is the smallest odd number of sides for which a regular polygon cannot be constructed with compass and straightedge. Understanding the undecagon helps solidify the general patterns that connect the number of sides to angle sums, diagonals, and symmetry.

Common Mistakes

Mistake: Confusing an undecagon (11 sides) with a dodecagon (12 sides) because the prefixes sound similar.
Correction: Remember that 'undeca-' (Latin) means eleven and 'dodeca-' (Greek) means twelve. Associating 'un-' with 'one more than ten' can help.
Mistake: Assuming each interior angle of a regular undecagon is a whole number of degrees.
Correction: The sum 1620° divided by 11 gives approximately 147.27°, which is not an integer. Always perform the division rather than rounding prematurely.

Related Terms

  • Polygon — General category that includes the undecagon
  • Side of a Polygon — An undecagon has exactly eleven sides
  • Regular Polygon — A regular undecagon has equal sides and angles
  • Decagon — Polygon with 10 sides, one fewer than undecagon
  • Dodecagon — Polygon with 12 sides, one more than undecagon
  • Diagonal — An undecagon has 44 diagonals
  • Interior Angle — Key measurement for any undecagon