A constructible polygon is a polygon whose vertices can be obtained by a compass-and-straightedge construction from the specified initial data. The classical question for a regular polygon asks which numbers of sides permit an exact construction from a given side or circumscribed circle.
A constructible regular polygon is a regular polygon admitting a compass-and-straightedge construction. The Gauss-Wantzel theorem classifies its possible numbers of sides.
For , a regular polygon with sides is a constructible regular polygon exactly when , where is a nonnegative integer and the are distinct Fermat primes; the product may be empty. Thus seventeen and fifteen qualify, while nine does not. Carl Friedrich Gauss established sufficiency, and Pierre Wantzel proved necessity in 1837.[ref]
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A constructible polygon is a polygon that can be drawn using only a compass and straightedge as per the principles of classical Greek geometry. Specifically, a regular polygon (one where all sides and angles are equal) is considered constructible if the number of its sides \( n \) can be expressed in a very specific way.