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.

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?

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

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

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

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. . . .

You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.

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

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

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .

The third of three articles on the History of Trigonometry.

The second of three articles on the History of Trigonometry.

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?

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

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

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

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

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.

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.

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