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?

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?

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?

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

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

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.

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?

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

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?

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

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.

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

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

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

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

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

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!

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

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

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

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.

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?

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.

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?

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

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?

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

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.

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

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?

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

A simple visual exploration into halving and doubling.

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

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

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. . . .

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

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?

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

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

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

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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

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

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