Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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

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?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

What is the same and what is different about these circle questions? What connections can you make?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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.

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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. . . .

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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

Can you find the area of a parallelogram defined by two vectors?

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

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

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Can you maximise the area available to a grazing goat?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

If you move the tiles around, can you make squares with different coloured edges?

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. . . .

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

How many winning lines can you make in a three-dimensional version of noughts and crosses?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Explore the effect of reflecting in two parallel mirror lines.

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Which of these games would you play to give yourself the best possible chance of winning a prize?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?