Constructing a 30° angle
Geometry construction using a compass and straightedge

This page shows how to construct (draw) a 30 degree angle with compass and straightedge or ruler. It works by first creating a rhombus and then a diagonal of that rhombus. Using the properties of a rhombus it can be shown that the angle created has a measure of 30 degrees. See the proof below for more on this.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

This construction works by creating a rhombus. Its two diagonals form four 30-60-90 triangles.

The image below is the final drawing above with the red items added.

  Argument Reason
1 Line segments PT, TR, RS, PS, TS are congruent (5 red lines) All created with the same compass width.
2 PTRS is a rhombus. A rhombus is a quadrilateral with four congruent sides.
3 Line segment AS is half the length of TS, and angle PAS is a right angle Diagonals of a rhombus bisect each other at right angles. See Rhombus definition.
4 Line segment AS is half the length of PS PS is congruent to TS. See (1), (3)
5 Triangle ∆PAS is a 30-60-90 triangle. ∆PAS is a right triangle with two sides in the ratio 1:2. (third side would be √3 by pythagoras).
6 Angle APS has a measure of 30°. In any triangle, smallest angle is opposite shortest side.

  - Q.E.D

Try it yourself

Click here for a printable worksheet containing two 30° angle exercises. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Constructions pages on this site

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Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

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