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. . . .

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

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.

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

The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.

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

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

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?

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.

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

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?

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?

'What Shape?' activity for adult and child. Can you ask good questions so you can work out which shape your partner has chosen?

Can you work out the area of the inner square and give an explanation of how you did it?

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. . . .

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

This article describes investigations that offer opportunities for children to think differently, and pose their own questions, about shapes.

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

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?

Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.

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

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

Use the isometric grid paper to find the different polygons.

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?

Use the interactivity to make this Islamic star and cross design. Can you produce a tessellation of regular octagons with two different types of triangle?

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

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

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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.

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?

Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

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

Can you each work out what shape you have part of on your card? What will the rest of it look like?

What shape is made when you fold using this crease pattern? Can you make a ring design?

Investigate these hexagons drawn from different sized equilateral triangles.

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

How many trapeziums, of various sizes, are hidden in this picture?

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.

What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

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?

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?