My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

How many centimetres of rope will I need to make another mat just like the one I have here?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

An investigation that gives you the opportunity to make and justify predictions.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

How many ways can you find of tiling the square patio, using square tiles of different sizes?

Here are many ideas for you to investigate - all linked with the number 2000.

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

Can you draw a square in which the perimeter is numerically equal to the area?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

I cut this square into two different shapes. What can you say about the relationship between them?

These practical challenges are all about making a 'tray' and covering it with paper.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

What do these two triangles have in common? How are they related?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

A follow-up activity to Tiles in the Garden.

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 shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

This article for teachers gives some food for thought when teaching ideas about area.

A simple visual exploration into halving and doubling.

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

Use the information on these cards to draw the shape that is being described.

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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

Look at the mathematics that is all around us - this circular window is a wonderful example.