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.

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?

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

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

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

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

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

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

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?

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?

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

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

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.

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?

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

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?

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?

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

A follow-up activity to Tiles in the Garden.

A simple visual exploration into halving and doubling.

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

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?

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?

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

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

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?

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!

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

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?

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?

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

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.

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

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

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?

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

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?

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

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

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