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?

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

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?

A follow-up activity to Tiles in the Garden.

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.

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 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).

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

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?

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

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

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.

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'.

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

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

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

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

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

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

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

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?

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?

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

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

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.

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?

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

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

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

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?

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. . . .

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

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

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. . . .

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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. . . .

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?

In how many ways can you halve a piece of A4 paper? How do you know they are halves?

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