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.

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?

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 .

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

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?

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

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?

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

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

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

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?

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. . . .

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

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

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

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

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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

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

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. . . .

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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.

There are lots of different methods to find out what the shapes are worth - how many can you find?

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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

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.

Can you describe this route to infinity? Where will the arrows take you next?