To construct an angle of 45 ° , start by constructing a right angle. To construct a right angle use the circle theorem which says that a triangle inscribed in a circle, one edge of which is a diameter, is always a right angled triangle.

Therefore, start by drawing a circle with a compass. The centre of the circle will be at the position of the compass needle.

Draw a diameter of the circle. Put the compass needle on one end of this diameter and draw an arc which crosses the circle at some point.

This point will be a vertex of the triangle. The other two vertices are the ends of the diameter. Join up the vertices. The vertex where the arc cuts the circle is the right angle.

The angle at A is a right angle, and bisecting this will give an angle of 45 ° . Put the compass point at A and draw an arc which crosses the other two sides of the triangle.

From where the arc crosses the sides of the triangle, without altering the compass length, draw extra arcs.

Draw a line from the right angle at A to the point where the arcs cross. This line will bisect the right angle, resulting in two angles of 45 ° each.