In how many ways can you halve a piece of A4 paper? How do you know they are halves?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
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 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?
How would you move the bands on the pegboard to alter these shapes?
What fractions of the largest circle are the two shaded regions?
Use the information on these cards to draw the shape that is being described.
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
This article for teachers gives some food for thought when teaching ideas about 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?
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?
Can you draw a square in which the perimeter is numerically equal to the area?
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.
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.
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 tiles do we need to tile these patios?
A simple visual exploration into halving and doubling.
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?
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?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
Look at the mathematics that is all around us - this circular window is a wonderful example.
Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?
What do these two triangles have in common? How are they related?
How many centimetres of rope will I need to make another mat just like the one I have here?
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).
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Derive a formula for finding the area of any kite.
Determine the total shaded area of the 'kissing triangles'.
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
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. . . .
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?
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?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
Can you work out the area of the inner square and give an explanation of how you did it?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
A task which depends on members of the group noticing the needs of others and responding.
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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 the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.