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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
An investigation that gives you the opportunity to make and justify
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
How many tiles do we need to tile these patios?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
How many centimetres of rope will I need to make another mat just
like the one I have here?
How would you move the bands on the pegboard to alter these shapes?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its 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?
A simple visual exploration into halving and doubling.
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?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
These practical challenges are all about making a 'tray' and covering it with paper.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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.
Here are many ideas for you to investigate - all linked with the
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
This article for teachers gives some food for thought when teaching
ideas about area.
Use the information on these cards to draw the shape that is being described.
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?
I cut this square into two different shapes. What can you say about
the relationship between them?
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.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Explore one of these five pictures.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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
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.
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. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
What do these two triangles have in common? How are they related?
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
A follow-up activity to Tiles in the Garden.
Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!
Can you work out the area of the inner square and give an
explanation of how you did it?