I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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.
Use the information on these cards to draw the shape that is being described.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
What do these two triangles have in common? How are they related?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
Determine the total shaded area of the 'kissing triangles'.
Derive a formula for finding the area of any kite.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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
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
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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
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.
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.
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
How would you move the bands on the pegboard to alter these shapes?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
An investigation that gives you the opportunity to make and justify
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
Can you work out the area of the inner square and give an
explanation of how you did it?
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 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 practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
A follow-up activity to Tiles in the Garden.
This article for teachers gives some food for thought when teaching
ideas about area.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
How many tiles do we need to tile these patios?
Explore one of these five pictures.
What happens to the area and volume of 2D and 3D shapes when you