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

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

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.

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

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

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.

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

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 red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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.

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

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

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

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

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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?

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

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?

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

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

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?

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

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.

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. . . .

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

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?

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?

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

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

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

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?

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. . . .

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?

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

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?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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

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

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 simple visual exploration into halving and doubling.

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

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

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