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 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.

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 .

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

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?

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?

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

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?

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

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?

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

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

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?

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?

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

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.

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

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

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

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

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?

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

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

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

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

Explore the effect of combining enlargements.

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

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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?

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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

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?

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.

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

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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?

Which set of numbers that add to 10 have the largest product?

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.

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.