Area and Perimeter of Plane Figures
271. The Circle's Boundary · Unwrapping the mystery of 2πr
The boundary length is always 2πr.
The circumference (boundary length) of a circle is 2πr, where r is the radius and π ≈ 3.14159 is the constant ratio of any circle's circumference to its diameter.
What this lesson covers
Try to break it
Drag P along the boundary to trace one full lap — the circumference. Drag R to change the radius. The circumference always equals 2πr — when r doubles, the boundary length doubles too. Try to find a circle whose boundary doesn't follow 2πr; impossible.
How you build it
Mark the centre O, draw the circle, mark a point P on its boundary, and join the radius OP.
- Point tool: mark the centre O near the middle of the canvas.
- Circle tool: click O, then click outward to set the radius and draw the circle.
- Point tool: click on the circle's boundary to place P exactly on it.
- Segment tool: join O to P — the radius r. The boundary length (circumference) is 2πr.
The proof, step by step
Prove that the circumference of a circle is 2πr.
- The ratio of circumference to diameter is constant for all circles.
- This constant ratio is denoted by π (pi).
- Since diameter d = 2r, we have C/d = π ⇒ C = πd = 2πr.
Worked example
The circumference of a circle is 44 cm. Find its radius. (Take π = 22/7)
Using C = 2πr, we get 44 = 2 × (22/7) × r. Solving for r gives r = 7 cm.
- 7 cm — correct
- 14 cm
- 3.5 cm
- 21 cm