This problem is intended to get children to look really hard at something they will see many times in the next few months.

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

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

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

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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

What is the same and what is different about these circle questions? What connections can you make?

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.

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

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

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

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.

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?

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?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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?

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.

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. . . .

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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

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 spot circles, spirals and other types of curves in these photos?

Investigate constructible images which contain rational areas.

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?

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

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

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

What do you think is the same about these two Logic Blocks? What others do you think go with them in the set?

This activity challenges you to make collections of shapes. Can you give your collection a name?

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

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 spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

Investigate these hexagons drawn from different sized equilateral triangles.

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. . . .

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

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

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

Can you help the children in Mrs Trimmer's class make different shapes out of a loop of string?

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

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?

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?

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