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.
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