How to construct (draw) a 45 degree angle with compass and straightedge or ruler

Constructing a 45° angle

Geometry construction using a compass and straightedge

Step-by-step Instructions

After doing this	Your work should look like this
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
2. Set the compass width to just over half the length of the line segment PQ.
3. With the compass point on P then Q, draw two arcs that cross above and below the line.
4. Draw a line between the two arc intersections. This is at right angles to PQ and bisects it (divides it in exactly half).

5. With the compass point on the intersection of PQ and the perpendicular just drawn, set the compass width to P
6. Draw an arc across the perpendicular, creating the point C
7. Draw a line from P through C, and on a little more. The end of this line is point R
8. Done. The angle ∠QPR has a measure of 45°

Proof

This construction works by creating an isosceles right triangle, which is a 45-45-90 triangle. The image below is the final drawing above with the red items added.

	Argument	Reason
1	Line segment AB is perpendicular to PQ.	Constructed that way. See Constructing the perpendicular bisector of a line.
2	Triangle APC is a right triangle	Angle ACP is 90° (from step 1)
3	Line segments CP,CA are congruent	Drawn with same compass width
4	Triangle ∆APC is isosceles.	CP = AC
5	Angle APC has a measure of 45°.	In isosceles triangle APC, base angles CPA and CAP are congruent. (See Isosceles Triangles). The third angle ACP is 90° and the interior angles of a triangle always add to 180. So both base angles CPA and CAP are 45°.

- Q.E.D

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.

Proof

Constructions pages on this site

Lines

Angles

Triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions