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There are **18** NRICH Mathematical resources connected to **Circumference and arc length**, you may find related items under Measuring and calculating with units.

Problem
Primary curriculum
Secondary curriculum
### Track Design

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

Problem
Primary curriculum
Secondary curriculum
### Efficient Cutting

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rollin' Rollin' Rollin'

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 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangles and Petals

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

Problem
Primary curriculum
Secondary curriculum
### Belt

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Arclets

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

Problem
Primary curriculum
Secondary curriculum
### Approximating Pi

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 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Over the Pole

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Flight Path

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Air Routes

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.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### A Rolling Disc - Periodic Motion

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Watching the Wheels Go 'round and 'round

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Five Circuits, Seven Spins

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Holly

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 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Illusion

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 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Contact

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

Problem
Primary curriculum
Secondary curriculum
### Giant Holly Leaf

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 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Just Rolling Round

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