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.
This article for teachers gives some food for thought when teaching ideas about area.
How would you move the bands on the pegboard to alter these shapes?
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.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
Use the information on these cards to draw the shape that is being described.
Look at the mathematics that is all around us - this circular window is a wonderful example.
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?
How many centimetres of rope will I need to make another mat just like the one I have here?
Derive a formula for finding the area of any kite.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
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?
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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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?
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?
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?
A simple visual exploration into halving and doubling.
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
How many ways can you find of tiling the square patio, using square tiles 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?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
Measure problems at primary level that require careful consideration.
Measure problems for primary learners to work on with others.
Measure problems for inquiring primary learners.
Measure problems at primary level that may require resilience.
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?
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?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
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.
An investigation that gives you the opportunity to make and justify predictions.
I cut this square into two different shapes. What can you say about the relationship between them?
Can you work out the area of the inner square and give an explanation of how you did it?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
These practical challenges are all about making a 'tray' and covering it with paper.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
How many tiles do we need to tile these patios?
A task which depends on members of the group noticing the needs of others and responding.
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?
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.
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.
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.