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.

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

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.

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?

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

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

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

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

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

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

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?

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

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

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

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?

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

A follow-up activity to Tiles in the Garden.

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

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

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

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

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?

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?

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

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

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?

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?

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?

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?

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

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!

A task which depends on members of the group noticing the needs of others and responding.

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

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

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

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

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?