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

Can you sort these triangles into three different families and explain how you did it?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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

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

The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?

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

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?

Can you explain why it is impossible to construct this triangle?

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

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?

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

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

Prove Pythagoras' Theorem for right-angled spherical triangles.

Dante, Jack and Jemma all solved this problem. Jamie even sent in a challenge of his own!

Simon and Andrei used different methods to compare the sizes of functions.

A description of how to make the five Platonic solids out of paper.

How do we measure curvature? Find out about curvature on soccer and rugby balls and on surfaces of negative curvature like banana skins.

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.