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?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
An investigation that gives you the opportunity to make and justify predictions.
This article for teachers gives some food for thought when teaching ideas about 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.
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.
How would you move the bands on the pegboard to alter these shapes?
Look at the mathematics that is all around us - this circular window is a wonderful example.
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?
Use the information on these cards to draw the shape that is being described.
A simple visual exploration into halving and doubling.
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
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?
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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?
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?
Derive a formula for finding the area of any kite.
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?
A task which depends on members of the group noticing the needs of others and responding.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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?
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?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
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. . . .
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Determine the total shaded area of the 'kissing triangles'.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
Explore one of these five pictures.
How many tiles do we need to tile these patios?
What do these two triangles have in common? How are they related?
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?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?