The perimeter of a circle is called a special name other than perimeter: circumference. The circumference of a circle is the length of the curve that encloses that circle. A circle is defined by only two things: its center and its radius. Two circles with the same center and the same radius are the same circle. Therefore, the circumference of a circle must depend on one of these, or both. In fact, the circumference is dependent solely on the radius of a circle: circumference equals 2πr, where r denotes the length of the radius. Another way to state the formula is πd, where d denotes the length of the diameter of the circle, which is, of course, twice that of the radius. A clever way to remember the formula for circumference is with the sentence "See two pies run." This sentence corresponds to written version of the formula, C = 2πr.

Another way to think of the curve that encloses a circle is through the 360 degree arc of that curve. Thus, the circumference of a circle is the length of the 360 degree arc of that circle. Since we know that the circumference of a 360 degree arc is 2πr, where r is the length of the radius, we can calculate the length of various arcs of a circle, provided that we know the radius of such a circle. For example, the length of a 180 degree arc must be half the circumference of the circle, the product of pi and the radius. The length of any arc is equal to whatever fraction of a full rotation the arc spans multiplied by the circumference of the circle. A 45 degree arc, for example, spans one-eighth of a full rotation, and is therefore equal to one-eighth the circumference of that circle. The length of an arc of n degrees equals (n/360) times the circumference. Below these concepts are pictured.