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

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.

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

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

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

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?

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

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

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

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?

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

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?

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?

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

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.

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?

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.

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?

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

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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

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

Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.

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?

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

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

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.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

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?

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

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?

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

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

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

A follow-up activity to Tiles in the Garden.

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

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?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

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

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