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

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

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?

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.

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?

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?

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

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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?

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 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?

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

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?

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. . . .

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

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

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. . . .

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

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

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

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

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.

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

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

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

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

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

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

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?

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

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

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.

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 ?

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

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

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

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.

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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?

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?