Locate the center of your equilateral triangle. Draw a line from each vertex to the opposite side to form a 90-degree angle with that side. The intersection of the three lines is the center.
Place the point of your compass on the center and the pencil on one of the triangle's vertexes. Rotate the pencil around the center point to form a circle.
Make a right triangle within your triangle. Highlight the portion of a line you drew in Step 1 from the center point straight down to the base of your triangle. Highlight the line from that point in the base to a vertex. Highlight from that vertex back to the center point. This line completes the right triangle and is equal to the radius of your circle. It also bisects the angle, which, because an equilateral triangle always has three 60-degree angles, leaves you with a 30-degree angle.
Use trigonometry to figure out the radius. You should already know the lengths of the sides of your equilateral triangle. The line from the center point to the base cuts the base in half, so you can use the cosine function. The cosine function of an angle within a right triangle is equal to the length of the adjacent side divided by the length of the hypotenuse. The hypotenuse, in this case, is equal to the radius, and we know that the angle is 30 degrees. For our purposes, let the sides of the triangle equal 6 units, meaning that the adjacent side equals 3 units. Therefore, you may write the equation cosine(30) = 3/radius.
Rewrite the equation to find the radius. Multiply both sides of the equation by the radius, then divide both sides by cosine(30) to get the equation radius = 3/(cosine 30). Our radius equals 3.46 units.
Determine the circumference. The equation for circumference is 2 times pi times radius. In our example, that is 2 x 3.14 x 3.46, which equals 21.7.