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.
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?
I cut this square into two different shapes. What can you say about the relationship between them?
What do these two triangles have in common? How are they related?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
How would you move the bands on the pegboard to alter these shapes?
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.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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.
Use the information on these cards to draw the shape that is being described.
Can you draw a square in which the perimeter is numerically equal to the area?
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.
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
This article for teachers gives some food for thought when teaching ideas about area.
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?
Here are many ideas for you to investigate - all linked with the number 2000.
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 is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?
A follow-up activity to Tiles in the Garden.
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?
In how many ways can you halve a piece of A4 paper? How do you know they are halves?
How many tiles do we need to tile these patios?
Explore one of these five pictures.
Look at the mathematics that is all around us - this circular window is a wonderful example.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
How many centimetres of rope will I need to make another mat just like the one I have here?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
Derive a formula for finding the area of any kite.
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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 practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
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.
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?
Measure problems for inquiring primary learners.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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).
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?