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

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.

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?

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

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?

Can you fill in the empty boxes in the grid with the right shape and colour?

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.

Use the isometric grid paper to find the different polygons.

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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.

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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 circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

How could you find out the area of a circle? Take a look at these ways.

Shogi tiles can form interesting shapes and patterns... I wonder whether they fit together to make a ring?

Take a look at the photos of tiles at a school in Gibraltar. What questions can you ask about them?

What shape and size of drinks mat is best for flipping and catching?

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 find the link between these beautiful circle patterns and Farey Sequences?

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

An environment that enables you to investigate tessellations of regular polygons

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

Two polygons fit together so that the exterior angle at each end of their shared side is 81 degrees. If both shapes now have to be regular could the angle still be 81 degrees?

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?

This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons.

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

Can you reproduce the Yin Yang symbol using a pair of compasses?

Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see?

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