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?

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

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?

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?

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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?

Use the isometric grid paper to find the different polygons.

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.

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?

Investigate these hexagons drawn from different sized equilateral triangles.

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 cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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.

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

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

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

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

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

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?

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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?

In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.

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. . . .

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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.

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

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.

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 LOGO challenge starts by looking at 10-sided polygons then generalises the findings to any polygon, putting particular emphasis on external angles

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

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.

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

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

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

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

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. . . .

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

An environment that enables you to investigate tessellations of regular polygons

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

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

What fractions of the largest circle are the two shaded regions?

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

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.

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?

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

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

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