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?

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

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

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 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.

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?

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?

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

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

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

A follow-up activity to Tiles in the Garden.

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

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

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

A simple visual exploration into halving and doubling.

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?

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.

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?

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?

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

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

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.

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.

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

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 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?

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

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

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

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

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

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

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

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

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

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).

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. . . .

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

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

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.

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?