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?

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

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

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

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

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

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?

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

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

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?

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

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

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?

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?

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.

A simple visual exploration into halving and doubling.

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

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.

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?

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

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

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

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

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?

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. . . .

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

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?

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?

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?

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

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

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

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

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?

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?

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

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

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

A follow-up activity to Tiles in the Garden.

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