Circumference and arc length

problem

Illusion

Age

11 to 16

Challenge level

A security camera, taking pictures each half a second, films a cyclist going by. In the film, the cyclist appears to go forward while the wheels appear to go backwards. Why?
problem

Watching the Wheels Go 'round and 'round

Age

7 to 11

Challenge level

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.
problem

Contact

Age

14 to 16

Challenge level

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?
problem

Rolling Inside

Age

14 to 16

Challenge level

Weekly Problem 11 - 2007
A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression for the number of revolutions the circle makes.
problem

Just Rolling Round

Age

14 to 16

Challenge level

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?
problem

Giant Holly Leaf

Age

14 to 16

Challenge level

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).
problem

Roll On

Age

14 to 16

Challenge level

Weekly Problem 5 - 2006
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?
problem

Running Race

Age

14 to 16

Challenge level

Weekly Problem 13 - 2006
If three runners run at the same constant speed around the race tracks, in which order do they finish?
problem

Five Circuits, Seven Spins

Age

16 to 18

Challenge level

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.
problem

Over the Pole

Age

16 to 18

Challenge level

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.