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

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?

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?

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

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

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

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. . . .

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

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

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?

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?

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

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

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

How many solutions can you find to this sum? Each of the different letters stands for a different number.

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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?

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

Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this. . . .

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

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?

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?

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

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

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

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

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

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

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. . . .

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.

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

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

A jigsaw where pieces only go together if the fractions are equivalent.

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

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?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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?

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

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