Circumference and arc length

problem
Favourite

Efficient Cutting

Age

11 to 14

Challenge level

3 out of 3

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.
problem
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Track Design

Age

14 to 16

Challenge level

1 out of 3

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

Arclets

Age

14 to 16

Challenge level

2 out of 3

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".
problem
Favourite

Triangles and Petals

Age

14 to 16

Challenge level

2 out of 3

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

Approximating Pi

Age

14 to 18

Challenge level

3 out of 3

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?
problem
Favourite

Belt

Age

16 to 18

Challenge level

1 out of 3

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
problem

Holly

Age

14 to 16

Challenge level

2 out of 3

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.
problem

Air Routes

Age

16 to 18

Challenge level

2 out of 3

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.
problem

Flight Path

Age

16 to 18

Challenge level

2 out of 3

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.
problem

Illusion

Age

11 to 16

Challenge level

3 out of 3

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?