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?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
An investigation that gives you the opportunity to make and justify
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 is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Here are many ideas for you to investigate - all linked with the
This article for teachers gives some food for thought when teaching
ideas about area.
These practical challenges are all about making a 'tray' and covering it with paper.
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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
A simple visual exploration into halving and doubling.
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?
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 many tiles do we need to tile these patios?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
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.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
I cut this square into two different shapes. What can you say about
the relationship between them?
Use the information on these cards to draw the shape that is being described.
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 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?
A follow-up activity to Tiles in the Garden.
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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
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
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
What do these two triangles have in common? How are they related?
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 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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you work out the area of the inner square and give an
explanation of how you did it?
Explore one of these five pictures.