Dividing a Circle into Thirds

Circles are everywhere in nature, art and the sciences. The sun and the moon, through spherical, form circles in the sky and travel in roughly circular orbits; the hands of a clock and the wheels on automobiles trace out circular paths; philosophically-minded observers speak of a the “circle of life.”

Circles in plain terms are mathematical constructs. You may need to know, using math, how to separate a complete circle into equal portions for pie, land or artistic purposes. If you have a pencil, along with a protractor, a compass or both, dividing a circle into three equal parts is straightforward and instructive.

A circle encloses 360 degrees of an arc, so for this exercise you need to create a “pie” with three equal 120° angles at the center.

Step 1: Draw the Diameter

Use your straightedge (ruler or protractor) to draw a diameter or line through the middle of the circle that reaches both edges. This of course divides your circle in half.

Step 2: Mark the Center

If the center of the circle is not marked, you will find it in this step because the diameter of any circle is the longest distance across the circle. Simply divide the value of the diameter by 2 and place a point halfway along the line from one edge to indicate the center.

Step 2: Measure Halfway to One Edge

Use your ruler or protractor to find a point exactly halfway between the center and one edge, or equivalently, one-fourth of the diameter or half of the radius. Label this point A.

Step 3: Draw a Perpendicular Line Through Point A to Both Edges

Use your protractor, or if necessary the short edge of your ruler, to draw a line through point A. Extend this line to the edges of the circle. Label the points at which this line intersects the edge of the circle B and C.

Step 4: Draw Lines from the Center to Points B and C

Using your straightedge, create lines connecting the center of the circle to points B and C. These lines represent radii of the circle, which have a value of half of the diameter.

Step 5: Use Geometry to Solve the Problem

You now have two right triangles inscribed within the circle. Because the short leg of each of these is one-half the distance of the hypotenuse of the circle, which is the same as a radius, you may recognize that these right triangles are “30-60-90” triangles, which have the property of the shortest side being half the length of the longest.

Because of this, you can conclude that the interior angles of the circle you have created between the two hypotenuses, and the hypotenuse and the diameter on the opposite side of the circle, are each 120°. You thus have a circle divided into three equal parts.