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.

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

Can you work out the fraction of the original triangle that is covered by the inner triangle?

Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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?

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. . . .

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Move the point P to see how P' moves. Then use your insights to calculate a missing length.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

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

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

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

A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

Four jewellers share their stock. Can you work out the relative values of their gems?

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 buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Can you find an efficent way to mix paints in any ratio?

Can you work out how to produce different shades of pink paint?

What's the most efficient proportion for a 1 litre tin of paint?

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

How long will it take Mary and Nigel to wash an elephant if they work together?

Is it cheaper to cook a meal from scratch or to buy a ready meal? What difference does the number of people you're cooking for make?

Which exact dilution ratios can you make using only 2 dilutions?

Which dilutions can you make using only 10ml pipettes?

Build a scaffold out of drinking-straws to support a cup of water