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Measuring and calculating with units > Circumference and arc length

##### 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?

##### Age 14 to 16 Challenge Level:

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

##### Age 14 to 16 Challenge Level:

The ten arcs forming the edges of the "holly leaf" are all arcs of
circles of radius 1 cm. Find the length of the perimeter of the
holly leaf and the area of its surface.

##### Age 11 to 14 Challenge Level:

Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

##### Age 14 to 16 Challenge Level:

An equilateral triangle rotates around regular polygons and
produces an outline like a flower. What are the perimeters of the
different flowers?

##### Age 14 to 16 Challenge Level:

Where should runners start the 200m race so that they have all run the same distance by the finish?

##### 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?

##### 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?

##### 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).

##### Age 14 to 18 Challenge Level:

By inscribing a circle in a square and then a square in a circle
find an approximation to pi. By using a hexagon, can you improve on
the approximation?

##### Age 14 to 16 Challenge Level:

A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .

##### Age 11 to 14 Challenge Level:

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.