Circumference is the distance around a circle. The formulas are C = pi * d (pi times the diameter) or equivalently C = 2 * pi * r (two times pi times the radius). The constant pi (approximately 3.14159) is the ratio of any circle's circumference to its diameter — this ratio is always the same regardless of the circle's size. Pi is irrational, meaning its decimal representation never terminates or repeats. Understanding circumference introduces students to this fundamental constant and to the geometry of curved shapes.
Have students measure the circumference and diameter of several circular objects (cans, plates, wheels) and compute the ratio — they will discover it is always approximately 3.14. This builds ownership of the concept rather than imposing the formula. Practice computing circumference given radius or diameter. Emphasize the relationship between radius and diameter (d = 2r). Use pi = 3.14 or the pi key on a calculator.
Circumference is simply the distance you would walk if you followed the edge of a circle all the way around. If you unwrapped that edge and laid it flat, it would form a straight line segment — and that length is the circumference. The question is: how long is that line compared to the circle's size?
The key discovery is that the ratio of circumference to diameter is always the same, no matter how big or small the circle. A bicycle wheel three feet across has a circumference exactly pi times three feet. A coin a quarter-inch across has a circumference exactly pi times a quarter inch. The constant pi (approximately 3.14159) is built into the geometry of circles — it is not something humans invented or chose, but a fixed fact of mathematics. This is why pi appears in both formulas: C = pi × d (using diameter) and C = 2 × pi × r (using radius). Since the diameter is always twice the radius (d = 2r), these two formulas say exactly the same thing.
To use the formulas, you just need to know which measurement you have. If you know the diameter, multiply by pi. If you know the radius, multiply by 2 and then by pi. For example, a circle with radius 5 cm has circumference C = 2 × pi × 5 = 10pi ≈ 31.4 cm. A circle with diameter 10 cm gives C = pi × 10 = 10pi — the same answer, because a radius of 5 cm means a diameter of 10 cm.
Pi is an irrational number, which means its decimal expansion goes on forever without repeating: 3.14159265358979... Using 3.14 is an approximation that is good enough for most purposes. When a problem asks for an exact answer, leave it in terms of pi (write "10pi" rather than "31.4"). When a problem asks for a decimal approximation, use 3.14 or the pi key on your calculator. These two circumference formulas are the foundation for more advanced circle geometry — the area formula A = pi × r² uses the same ingredients, and arc length (a portion of circumference) extends the same idea to parts of circles.