What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

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?

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

Can you explain what is happening and account for the values being displayed?

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?

The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

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?

There are many different methods to solve this geometrical problem - how many can you find?

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

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 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.

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

How can you represent the curvature of a cylinder on a flat piece of paper?

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.

What is an AC voltage? How much power does an AC power source supply?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.