What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

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?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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

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

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

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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

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

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.

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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. . . .

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

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

How many centimetres of rope will I need to make another mat just like the one I have here?

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

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.

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

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

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

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?

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

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

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?

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?

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

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

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.

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

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?

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

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

A follow-up activity to Tiles in the Garden.

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

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?

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?

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?

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

What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.