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 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 area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Describe how to construct three circles which have areas in the ratio 1:2:3.

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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.

If you were to set the X weight to 2 what do you think the angle might be?

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

The second of three articles on the History of Trigonometry.

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

The first of three articles on the History of Trigonometry. This takes us from the Egyptians to early work on trigonometry in China.

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

How far should the roof overhang to shade windows from the mid-day sun?

The third of three articles on the History of Trigonometry.

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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

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 ?

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

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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 problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

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.

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

How would you design the tiering of seats in a stadium so that all spectators have a good view?