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

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. . . .

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

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?

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?

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

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

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

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

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

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

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?

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

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

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

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

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

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

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.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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?

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

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!

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

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

A follow-up activity to Tiles in the Garden.

A simple visual exploration into halving and doubling.

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?

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?

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 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?

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?

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

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

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

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?

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?

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

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.

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