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?

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

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

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

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

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

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

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?

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

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.

How would you move the bands on the pegboard to alter these shapes?

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

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?

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

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?

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?

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

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

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.

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

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

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

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?

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?

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

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?

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

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?

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

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

A simple visual exploration into halving and doubling.

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

In how many ways can you halve a piece of A4 paper? How do you know they are halves?

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

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

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

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

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 work out the area of the inner square and give an explanation of how you did it?

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?

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

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

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

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

Determine the total shaded area of the 'kissing triangles'.

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?

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

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.