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?

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?

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

A follow-up activity to Tiles in the Garden.

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?

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

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.

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?

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

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

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.

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

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

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?

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

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

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?

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

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.

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

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

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

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

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. . . .

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.

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?

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!

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

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?

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).

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .

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

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

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?

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

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?

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

A simple visual exploration into halving and doubling.

What fractions of the largest circle are the two shaded regions?

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

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

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.