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

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.

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

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

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

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

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

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.

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

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

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?

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

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

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?

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.

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

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

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

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 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?

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

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?

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

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?

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

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?

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

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.

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

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?

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.

A simple visual exploration into halving and doubling.

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

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

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

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

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

A follow-up activity to Tiles in the Garden.

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

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

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. . . .

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

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?