Circumference and arc length

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Favourite

Approximating Pi

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?
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Rollin' Rollin' Rollin'

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?
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Efficient Cutting

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.
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Track Design

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?
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Favourite

Arclets

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".
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Favourite

Triangles and Petals

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?
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Favourite

Belt

Age

16 to 18

Challenge level

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.
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Holly

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.
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Air Routes

Age

16 to 18

Challenge level

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

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.