Constructible Angles

To contact us Click HERE


A student asked what angles are constructible, consideringonly angles that measure a whole number of degrees (which I'll call integralangles). The answer is very simple: The only constructible integral angles arethose which measure 3n degrees, where n is any natural number.
To see this, first note that angles of measure 60 and 72(vertex angles in a regular pentagon) are constructible. Since angles may bebisected repeatedly by construction, this means that angles of measure 15 =60/4 and 9 = 72/8 degrees may be constructed. Since 3 = 9*2 – 15, an angle of 3degrees may be constructed by constructing two adjacent 9 degree angles, andthen a 15 degree angle inside the resulting 18 degree angle. By adding n 3-degreeangles together, a 3n degree angle can be constructed.
The proof that only these integral angles can be constructedseems to require the more advanced result that there is at least one integralangle that is not constructible. For example, an angle of 20 degrees isnon-constructible. (This is the standard proof, using Galois theory, thatdemonstrates the impossibility of angle trisection.) If there were an integralangle of measure n that is not a multiple of 3 that is constructible, thensince (n,3) = 1, there is a linear combination with integral coefficients of nand 3 that gives 1, and so an angle of 1 degree would be constructible, andhence an angle of 20 degrees would be constructible, which it is not..