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 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?

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

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

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 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.

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

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

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

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

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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. . . .

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 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?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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?

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.

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?

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

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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

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?

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

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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

Explore the effect of reflecting in two parallel mirror lines.

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. . . .

Explore the effect of combining enlargements.

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .

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

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?

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?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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.

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?

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

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

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 ?

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

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