You can make a simple "makeshift" compass using nothing but a piece of string, a pencil and a thumbtack.
Determine how large you want your circle to be. Cut a piece of string that is about 8 inches longer than the diameter of your circle.
Tie a slipknot on both ends of the string. Make sure the distance between the knots is half as long as the total distance across your circle. The distance between the knots will be the radius (half the diameter) of your circle.
Secure the pencil in the slipknot on one end of the string.
Secure the thumbtack in the slipknot on the other end of the string.
Place the tack on the very center of where you want your circle. You do not need to push the tack into anything. The sharp point on the tack will keep it from slipping with just with a small amount of pressure applied.
Hold the tack while pulling the pencil out to make the string tight. If the size of your circle needs to be exact, make a mark with the pencil and then use a ruler to measure the distance between the center dot and the mark you just made. Make any adjustments necessary in the length of the string.
Draw the circle with the pencil, keeping the string tight at all times and the pencil straight up and down. (Tilting the pencil different ways while drawing will make the circle slightly imperfect in places.)
You can use this basic method to make very large circles, even 100 feet in diameter. Just use a thin rope instead of string and a thin tent stake or sharpened dowel rod instead of a thumbtack. Replace the pencil with whatever will work best on the surface you are working on. (Try chalk for pavement and sprinkled lime or spray paint for grass.)