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?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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

Can you find a relationship between the area of the crescents and the area of the triangle?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

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.

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

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?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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?

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.

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?

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. . . .

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

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.

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.

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.

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

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?

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.

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?

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .

Can you find the areas of the trapezia in this sequence?

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

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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!

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

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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

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 is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Where should runners start the 200m race so that they have all run the same distance by the finish?

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.

Investigate constructible images which contain rational areas.

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.