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.

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.

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. . . .

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

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

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

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 simple visual exploration into halving and doubling.

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

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

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?

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

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

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?

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?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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

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

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?

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

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?

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

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?

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

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

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

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?

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?

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.

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

A follow-up activity to Tiles in the Garden.

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

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?

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

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?

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

This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?

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

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

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?

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

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

What happens to the area and volume of 2D and 3D shapes when you enlarge them?