constructible angles with integer values in degrees

The aim is to characterize all constructible angles with straightedge and compass whose value is an integer number of degrees (like 60 or 36). From now on, every time we refer to the measurement of an angle, it is meant to be in degrees, not radians.

We need two short lemmas:

Lemma 1

If an angle measuring x degrees can be constructed, then angles measuring


can be constructed.

Notice that we are not stating all of them have integer values, only constructibility. The proof follows almost inmediately by knowing any angle can be bisected with ruler and compass.

Lemma 2

If an angle measuring x degrees can be constructed, then angles measuring any integer multipleMathworldPlanetmath of x, that is, 2x,3x,4x, can be constructed

If you can construct x, you can construct again an adjacentPlanetmathPlanetmathPlanetmath angle with the same value and you will have constructed an angle measuring 2x. Repeat the procedure and you get 3x,4x,.

Now, a theorem.

Theorem 1

The angle measuring 3 can be constructed.

It is well known that both regular pentagon and equilateral triangleMathworldPlanetmath can be built with ruler and compass. That allows us to construct angles measuring 72 and 60.

By first lemma we can construct then

72,722=36,362=18,182=9,92=4.5=4 30

and also we can construct

60,602=30,302=15,152=7.5=7 30

But if we can construct 4 30 and 7 30 we can then construct their difference, which is exactly 3.

Alternative (J. Pahikkala): Since 72 and 60 can be constructed, 12=72-60 can be also constructed. Bisecting 12 gives 6 and bisecting again shows that 3 can be constructed.

Theorem 2

We can construct any angle measuring an integer multiple of 3.

The proof follows directly from the second Lemma.

Theorem 3

The only constructible angles measuring an integer number of degrees are precisely the multiples of 3.

We are only left to prove we cannot construct any other integer value. We will work by contradiction.

Suppose we are able to construct with ruler and compass an angle measuring t with t integer and t not multiple of 3.

Since 3 does not divide t and 3 is prime, it follows that 3 and t are coprimeMathworldPlanetmath, that is, gcd(3,t)=1.

But then, by Euclid’s algorithm we can find integers m,n so that 3m-tn=1 (n or m could be negative).

By the second lemma, we can construct both 3m and tn, so we can construct their sum (or difference), which would prove 1 is constructible, and therefore any angle equal to an integer number of degrees could be constructed with ruler and compass.

However, the standard proof of the impossibility of trisecting an arbitrary angle goes by proving 20 cannot be constructed with ruler and compass, this contradicts what we just showed, and therefore only angles being an integer multiple of 3 can be constructed.


For a more general proof for other real values besides integers, see the theorem on constructible angles.

Title constructible angles with integer values in degrees
Canonical name ConstructibleAnglesWithIntegerValuesInDegrees
Date of creation 2013-03-22 14:16:36
Last modified on 2013-03-22 14:16:36
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 9
Author PrimeFan (13766)
Entry type Theorem
Classification msc 11S20
Classification msc 11R32
Classification msc 51M15
Classification msc 13B05
Synonym constructible angle
Related topic ExactTrigonometryTables
Related topic TheoremOnConstructibleAngles
Related topic ClassicalProblemsOfConstructibility