Mensuration
153. The Circle's Edge · always 2π times the radius
The straight line length always equals 2π times the radius.
Circumference of a circle = 2πr = πd (where d = 2r is the diameter). The ratio of any circle's circumference to its diameter is always π ≈ 3.14159. A foundational geometric constant.
What this lesson covers
Try to break it
Drag P to change the radius. The unrolled circumference (the straight line) always stretches to exactly 2π × r. Try to make it shorter or longer than 2πr — you can't, the unroll keeps it locked.
How you build it
Draw a circle and its circumference line.
- Mark the centre O of the circle.
- Mark point P somewhere on the canvas — OP will be the radius.
- With centre O and radius OP, draw the circle. Click O, then P.
- Draw segment OP — the radius r. The full circumference traced around the circle is exactly 2π × r, about 6.28 times this length.
The proof, step by step
Prove that the circumference of a circle is 2π × radius.
- Imagine bending a straight wire of length L into a perfect circle.
- The length of the wire becomes the circumference C of the circle.
- By definition, π is the ratio of circumference to diameter: π = C / (2r).
- Rearranging the formula gives the fundamental result: C = 2πr.
Worked example
A circular park has a radius of 35 m. Find its circumference. (Take π = 22/7)
C = 2πr = 2 × (22/7) × 35 = 220 m. The circumference is 220 m.
- 220 m — correct
- 110 m
- 440 m
- 70 m