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n-gon

n-gon

A polygon with n sides. For example, an 8-gon is another name for an octagon.

 

See also

Triangle, quadrilateral, pentagon, hexagon, heptagon, nonagon, decagon, undecagon, dodecagon

Key Formula

S=(n−2)×180°S = (n - 2) \times 180°
Where:
  • SS = Sum of the interior angles of the n-gon
  • nn = Number of sides (and angles) of the polygon

Worked Example

Problem: Find the sum of the interior angles and the measure of each interior angle of a regular 12-gon (dodecagon).
Step 1: Identify n. A 12-gon has 12 sides, so n = 12.
n=12n = 12
Step 2: Use the interior angle sum formula to find the total.
S=(12−2)×180°=10×180°=1800°S = (12 - 2) \times 180° = 10 \times 180° = 1800°
Step 3: Since the 12-gon is regular (all sides and angles equal), divide the total by the number of angles.
Each angle=1800°12=150°\text{Each angle} = \frac{1800°}{12} = 150°
Answer: The sum of the interior angles of a regular 12-gon is 1800°, and each interior angle measures 150°.

Another Example

Problem: A regular n-gon has each interior angle measuring 140°. How many sides does it have?
Step 1: In a regular n-gon, each interior angle equals the total angle sum divided by n.
(n−2)×180°n=140°\frac{(n - 2) \times 180°}{n} = 140°
Step 2: Multiply both sides by n and solve.
(n−2)×180=140n(n - 2) \times 180 = 140n
Step 3: Expand and isolate n.
180n−360=140n  ⟹  40n=360  ⟹  n=9180n - 360 = 140n \implies 40n = 360 \implies n = 9
Answer: The polygon is a 9-gon (nonagon).

Frequently Asked Questions

What is a 1000-gon or 100-gon called?
A 100-gon is called a hectogon, and a 1000-gon is called a chiliagon. In practice, mathematicians almost always just say '100-gon' or '1000-gon' because the Greek names become obscure at large values of n. The n-gon notation is preferred precisely because it works for any number.
What is the smallest possible n-gon?
The smallest n-gon is a 3-gon, which is a triangle. A polygon must have at least 3 sides to enclose a region of the plane, so n must be 3 or greater.

n-gon (general polygon) vs. Regular n-gon

An n-gon is any polygon with n sides — its sides and angles can have different measures. A regular n-gon is a special case where all n sides are equal in length and all n interior angles are equal in measure. For example, a rectangle is a 4-gon but not a regular 4-gon; a square is a regular 4-gon.

Why It Matters

The n-gon notation lets you state general formulas and theorems that apply to all polygons at once, regardless of the number of sides. Instead of proving angle-sum results separately for triangles, quadrilaterals, pentagons, and so on, you prove it once for the general n-gon. This notation also appears frequently in competition math, engineering, and computer graphics whenever polygons with large or variable numbers of sides are discussed.

Common Mistakes

Mistake: Thinking n can be 1 or 2, and trying to create a '1-gon' or '2-gon'.
Correction: A polygon must have at least 3 sides. Values of n below 3 do not form a closed figure with interior area, so the smallest n-gon is a triangle (n = 3).
Mistake: Assuming every n-gon is regular (all sides and angles equal).
Correction: The term n-gon refers to any polygon with n sides. It could be irregular with unequal sides and angles. You must say 'regular n-gon' if you mean all sides and angles are congruent.

Related Terms