theorem on constructible angles

Theorem 1.

Let θR. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    An angle of measure ( θ is constructiblePlanetmathPlanetmath (;

  2. 2.

    sinθ is a constructible number;

  3. 3.

    cosθ is a constructible number.


First of all, due to periodicity, we can restrict our attention to the intervalMathworldPlanetmathPlanetmath 0θ<2π. Even better, we can further restrict our attention to the interval 0θπ2 for the following reasons:

  1. 1.

    If an angle whose measure is θ is constructible, then so are angles whose measures are π-θ, π+θ, and 2π-θ;

  2. 2.

    If x is a constructible number, then so is |x|.

If θ{0,π2}, then clearly an angle of measure θ is constructible, and {sinθ,cosθ}={0,1}. Thus, equivalence ( has been established in the case that θ{0,π2}. Therefore, we can restrict our attention even further to the interval 0<θ<π2.

Assume that an angle of measure θ is constructible. Construct such an angle and mark off a line segmentMathworldPlanetmath of length 1 from the vertex ( of the angle. Label the endpoint that is not the vertex of the angle as A.


Drop the perpendicularPlanetmathPlanetmath from A to the other ray of the angle. Since the legs of the triangleMathworldPlanetmath are of lengths sinθ and cosθ, both of these are constructible numbers.


Now assume that sinθ is a constructible number. At one endpoint of a line segment of length sinθ, erect the perpendicular to the line segment.


From the other endpoint of the given line segment, draw an arc of a circle with radius 1 so that it intersects the erected perpendicular. Label this point of intersectionDlmfMathworld as A. Connect A to the endpoint of the line segment which was used to draw the arc. Then an angle of measure θ and a line segment of length cosθ have been constructed.


A similarMathworldPlanetmath procedure can be used given that cosθ is a constructible number to prove the other two statements. ∎

Note that, if cosθ0, then any of the three statements thus implies that tanθ is a constructible number. Moreover, if tanθ is constructible, then a right triangle having a leg of length 1 and another leg of length tanθ is constructible, which implies that the three listed conditions are true.

Title theorem on constructible angles
Canonical name TheoremOnConstructibleAngles
Date of creation 2013-03-22 17:15:59
Last modified on 2013-03-22 17:15:59
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 33B10
Classification msc 51M15
Classification msc 12D15
Related topic ConstructibleNumbers
Related topic CompassAndStraightedgeConstruction
Related topic ConstructibleAnglesWithIntegerValuesInDegrees
Related topic ExactTrigonometryTables
Related topic ClassicalProblemsOfConstructibility
Related topic CriterionForConstructibilityOfRegularPolygon