A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

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?

This article for pupils gives some examples of how circles have featured in people's lives for centuries.

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.

See if you can anticipate successive 'generations' of the two animals shown here.

Introducing a geometrical instrument with 3 basic capabilities.

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .

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.

In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.

Can you reproduce the design comprising a series of concentric circles? Test your understanding of the realtionship betwwn the circumference and diameter of a circle.

Thinking of circles as polygons with an infinite number of sides - but how does this help us with our understanding of the circumference of circle as pi x d? This challenge investigates. . . .

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

This interactivity allows you to sort logic blocks by dragging their images.

A metal puzzle which led to some mathematical questions.

What fractions of the largest circle are the two shaded regions?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.

The three corners of a triangle are sitting on a circle. The angles are called Angle A, Angle B and Angle C. The dot in the middle of the circle shows the centre. The counter is measuring the size. . . .

Look at the mathematics that is all around us - this circular window is a wonderful example.

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?

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?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Can Jo make a gym bag for her trainers from the piece of fabric she has?

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?

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?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?