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.
Look at the mathematics that is all around us - this circular window is a wonderful example.
How would you move the bands on the pegboard to alter these shapes?
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.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
A simple visual exploration into halving and doubling.
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?
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. . . .
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Use the information on these cards to draw the shape that is being described.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
Can you draw a square in which the perimeter is numerically equal to the area?
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
An investigation that gives you the opportunity to make and justify predictions.
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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
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?
Explore one of these five pictures.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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?
Can you work out the area of the inner square and give an explanation of how you did it?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Measure problems at primary level that require careful consideration.
Measure problems at primary level that may require determination.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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 do these two triangles have in common? How are they related?
Measure problems for inquiring primary learners.
A follow-up activity to Tiles in the Garden.
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
These practical challenges are all about making a 'tray' and covering it with paper.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
How many tiles do we need to tile these patios?
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?
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.
I cut this square into two different shapes. What can you say about the relationship between them?