We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point. . . .

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?

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

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.

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

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?

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?

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

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.

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

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

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. . . .

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.

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?

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.

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.

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.

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.

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.

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

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.

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

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.

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

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?

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

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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.

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of. . . .

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

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!

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. . . .

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

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

Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...

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.

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

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

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

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

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 farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

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.

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.