Q&A for How to Construct a 60 Degrees Angle Using Compass and Ruler

Draw a base line. Pick a point on the base line. Mark an equal distance from that point in each direction of the line so that you now have two new points on the base line. Use the compass to strike two arcs of equal radius, one centered on each of the two new points. Draw a line from the original point on the base line to the intersection(s) of the arcs. This new line is perpendicular to the base line and makes two 90-degree angles with it.

Pick a point on a line to be the vertex of your angle. Mark a convenient distance along the line from the vertex. Draw a new line perpendicular to the first line at the point you measured. Measure from that point along the perpendicular line a distance equal to the distance along the first line from the vertex to the base of the perpendicular line. Mark that last measurement. Draw a line from the vertex to that last mark you made. The new line makes a 45-degree angle with the original line.

Construct a 60° angle as shown above. Then bisect it using a compass and straightedge. You wind up with two 30° angles.

Construct a 90° angle and bisect it. (See the wikiHow article Construct a 90 Degrees Angle Using Compass and Ruler .) Bisect the angle this way: Strike an arc through both legs of the 90° angle. From each point of intersection (of the arc and legs), strike arcs of the same radius such that they intersect each other. From the vertex of the 90° angle, draw a straight line to a point of intersection between the two arcs. That line bisects the 90° angle, giving you two 45° angles.

Draw a line, and establish a point on that line as the vertex of the angle you're going to construct. With a compass, mark off a convenient distance from the vertex in each direction on the line, establishing two new points on the line. From each of those new points, strike an arc of the same radius -- a radius longer than the distance from either new point to the vertex. Those two arcs will intersect at points above and below the vertex. Draw a straight line connecting the two arc points of intersection. That line will be perpendicular to the original line and form four 90° angles with it.

An arc of a circle is an unbroken section of its perimeter, like the crust of a slice of pizza.

A 60-degree angle is one that is often encountered in geometric exercises. You find three such angles, for instance, in an equilateral triangle.

A 135° angle is the supplement of a 45° angle. (That means that a 45° angle and a 135° angle add together to make a 180° angle -- or a straight line.) So on a straight line, construct a perpendicular line using a compass, and bisect the 90° angle to get two 45° angles. The supplement of either of the 45° angles is a 135° angle.

Let's assume you don't have a protractor. First draw a 90° angle. (See Construct a 90 Degrees Angle Using Compass and Ruler .) Separately draw a 30° angle. (See Construct a 30 Degrees Angle Using Compass and Straightedge .) Bisect the 30° angle. (See Construct a Bisector of a Given Angle .) That gives you a 15° angle. Use a compass to transfer the 15° angle to one inside edge of the 90° angle you constructed earlier. The rest of the 90° angle consists of 75°.

The circumference is πd = 7π = 21.99 cm.

If you don't have a protractor, do this: draw a straight line. On it construct a 45° angle (as shown in an answer above). Bisect that angle with a compass. The resulting angle is 22½°. The supplement of 22½° is 157½°.

Use Archimedes' angle trisection.