This task looks at the different turns involved in different Olympic sports as a way of exploring the mathematics of turns and angles.

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

Can you describe the journey to each of the six places on these maps? How would you turn at each junction?

How much do you have to turn these dials by in order to unlock the safes?

Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Can you find triangles on a 9-point circle? Can you work out their angles?

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Draw some angles inside a rectangle. What do you notice? Can you prove it?

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

Geometry problems at primary level that may require resilience.

Geometry problems at primary level that require careful consideration.

Geometry problems for primary learners to work on with others.

Geometry problems for inquiring primary learners.

Join pentagons together edge to edge. Will they form a ring?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

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?

Make a clinometer and use it to help you estimate the heights of tall objects.

Make an equilateral triangle by folding paper and use it to make patterns of your own.

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

Can you draw perpendicular lines without using a protractor? Investigate how this is possible.

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?

Suggestions for worthwhile mathematical activity on the subject of angle measurement for all pupils.

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

Use your knowledge of angles to work out how many degrees the hour and minute hands of a clock travel through in different amounts of time.

How many right angles can you make using two sticks?

Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.

Have you ever noticed how mathematical ideas are often used in patterns that we see all around us? This article describes the life of Escher who was a passionate believer that maths and art can be. . . .

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

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?

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.