Heptadecagon — Definition, Formula & Examples

A heptadecagon is a polygon with 17 sides and 17 vertices. It is historically famous because Carl Friedrich Gauss proved at age 19 that a regular heptadecagon can be constructed using only a compass and straightedge.

A heptadecagon is a 17-gon, a closed plane figure composed of 17 straight line segments (sides) meeting at 17 vertices. A regular heptadecagon has all sides congruent and all interior angles congruent, with each interior angle measuring exactly $\frac{(17-2) \cdot 180°}{17} = \frac{2700°}{17} \approx 158.82°$ .

Key Formula

\text{Each interior angle} = \frac{(n-2) \cdot 180°}{n} = \frac{(17-2) \cdot 180°}{17} = \frac{2700°}{17}

Where:

$n$ = Number of sides, which equals 17 for a heptadecagon

Worked Example

Problem: Find the sum of interior angles and the number of diagonals of a heptadecagon.

Sum of interior angles: Use the formula for the sum of interior angles of any polygon.

(17 - 2) \times 180° = 15 \times 180° = 2700°

Number of diagonals: Use the diagonal formula for an n-sided polygon.

\frac{17(17-3)}{2} = \frac{17 \times 14}{2} = \frac{238}{2} = 119

Answer: A heptadecagon has an interior angle sum of 2700° and 119 diagonals.

Why It Matters

In 1796, Gauss's proof that the regular heptadecagon is constructible with compass and straightedge was a landmark result in mathematics. It connected polygon construction to number theory, specifically Fermat primes, and influenced Gauss's decision to pursue mathematics as a career. This result appears in courses on abstract algebra and the history of mathematics.

Common Mistakes

Mistake: Confusing "heptadecagon" (17 sides) with "heptagon" (7 sides) because both start with "hept."

Correction: The prefix "hepta" means 7, but "heptadeca" means 7 + 10 = 17. Similarly, a hexadecagon has 16 sides (6 + 10).