Circles and Circle Theorems - February 2004, All Stages

Problems

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Overlapping Circles

Age 7 to 11 Challenge Level:

What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?

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A Square in a Circle

Age 7 to 11 Challenge Level:

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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Sponge Sections

Age 7 to 11 Challenge Level:

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

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An Unusual Shape

Age 11 to 14 Challenge Level:

Can you maximise the area available to a grazing goat?

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Rolling Around

Age 11 to 14 Challenge Level:

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

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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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Maximised Area

Age 14 to 16 Short Challenge Level:

Weekly Problem 39 - 2011
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

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Annulus Area

Age 14 to 16 Short Challenge Level:

Weekly Problem 38 - 2011
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?

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Pencil Turning

Age 14 to 16 Short Challenge Level:

Weekly Problem 37 - 2011
Rotating a pencil twice about two different points gives surprising results...

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Circle in a Semicircle

Age 14 to 16 Short Challenge Level:

Weekly Problem 36 - 2011
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

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Circle Box

Age 14 to 16 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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Inscribed in a Circle

Age 14 to 16 Challenge Level:

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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Circle Scaling

Age 14 to 16 Challenge Level:

Describe how to construct three circles which have areas in the ratio 1:2:3.

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Circles in Circles

Age 16 to 18 Challenge Level:

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

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Lunar Angles

Age 16 to 18 Challenge Level:

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

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Flower

Age 16 to 18 Challenge Level:

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.