Constructing a 45° angle

This shows how to construct a 45° angle from scratch using a compass and straightedge. If you already have one of the angle sides given, you can skip the first step.

By constructing a 45° angle and then bisecting it you can make a 22.5° angle.
See Bisecting an angle with compass and straightedge

Also see:
Constructing a 30° angle
Constructing a 60° angle

Instructions Click on 'Next' to go through the construction one step at a time, or click on 'Run' to let it run without stopping.
(If there is no image below, see support page.)


45° is exactly half 90° - a right angle. So you can always do this by constructing a 90° angle and bisecting it. The method shown here has slightly fewer steps, but either approach is perfectly acceptable.
Step-by-step Instructions
Step 1 Draw a line segment which will become one side of the angle. (Skip this step if you are given this line.) The exact length is not important. Label it PQ. P will be the angle's vertex.
In the next 3 steps we create the perpendicular bisector of PQ. See Constructing a perpendicular bisector of a line segment
Step 2 Set the compass width to just over half the length of the line segment PQ.
Step 3 With the compass point on P then Q, draw two arcs that cross above and below the line.
Step 4 Draw a line between the two arc intersections. This is at right angles to PQ and bisects it (divides it in exactly half).
Step 5 With the compass point on the intersection of PQ and the perpendicular just drawn, set the compass width to P
Step 6 Draw an arc across the perpendicular, creating the point C
Step 7 Draw a line from P through C, and on a little more. The end of this line is point R
Step 8 Done. The angle QPR has a measure of 45°
Try it yourself
Click here for a printable worksheet containing two 45° 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.

Other constructions

Lines

Angles

Triangles

Triangle Centers

Circles, Arcs and Ellipses

Non-Euclidean constructions