In order to construct an equilateral triangle within a circle, you will need a compass, a straight edge, and a right angle drafting triangle. In lieu of a drafting triangle, any flat object with a true 90-degree angle, such as a sheet of paper, can be used.
- Find the diameter of the circle
Place the 90-degree corner of the drafting triangle, or any corner of the sheet of paper, at any point on the perimeter of the circle. Orient the triangle or the sheet of paper so that it crosses the perimeter at two other points. Mark those two points and draw a line connecting them using a straight edge. This line is a diameter of the circle.
- Find a second diameter
Choose a different point on the perimeter and find a second diameter using the same method as in Step 1. The point where these two diameters cross is the center of the circle.
- Find the radius of the circle
Set the distance on your compass so that it equals the distance from the center of the circle to the perimeter. This is the radius of the circle.
- Mark six equidistant points on the perimeter of the circle
With the distance on your compass still set equal to the radius of the circle, place the point of the compass anywhere on the perimeter of the circle and mark off a small arc on the circle. Move the point of the compass to the point where that arc crosses the circle and make another arc. Continue around the circle four more times. Your last mark should be in exactly the same place as your first mark.
- Connect every other dot
All six dots form a regular hexagon inscribed within the circle. Connect every other dot by drawing a line with a straight edge. The result is an equilateral triangle.