A Rolling Circle (The Path Traced by a Moving Circle)
When a circle rolls along a straight line or another circle, we consider the paths traced by the “center of the circle” and by a “point on the circumference.”
Key Points About a Rolling Circle
- Path of the center: When a circle rolls along a straight line, the center traces a parallel line at a distance equal to the radius from the surface.
- Path of a point on the circumference: It traces a curve called a cycloid (entrance examinations often ask about the length of this curve).
- Distance rolled = distance traveled by the center
Example 1: A circle with radius 3 cm rolls one complete revolution along a straight line. How many centimeters does the center move? (π = 3.14)
Solution:
① Circumference = 2 × 3 × 3.14 = 18.84 cm
② In one complete revolution, the center moves exactly one circumference.
③ Answer: 18.84 cm
Solution:
① Circumference = 2 × 3 × 3.14 = 18.84 cm
② In one complete revolution, the center moves exactly one circumference.
③ Answer: 18.84 cm
Example 2 (Rolling Around an L-Shaped Corner): A circle with radius 2 cm rolls around and turns at an outward 90° corner. What sector does the center trace while turning, and what is the length of its arc? (π=3.14)
Solution:
① The center traces a sector with radius 2 cm centered at the corner point.
② The angle is 90°.
③ Arc length = 2 × 3.14 × 2 × (90 ÷ 360) = 3.14 cm
Solution:
① The center traces a sector with radius 2 cm centered at the corner point.
② The angle is 90°.
③ Arc length = 2 × 3.14 × 2 × (90 ÷ 360) = 3.14 cm
When turning around a corner, the radius of the arc is often “the radius of the rolling circle + the radius of the corner,” so confirm it with a diagram.