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 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?

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?

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

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

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

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

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?

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?

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?

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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?

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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 all unit fractions be written as the sum of two unit fractions?

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

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?

The clues for this Sudoku are the product of the numbers in adjacent squares.

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

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

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

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

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

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?

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

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

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Use the differences to find the solution to this Sudoku.

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

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.

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

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

How many different symmetrical shapes can you make by shading triangles or squares?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?