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 is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
I cut this square into two different shapes. What can you say about
the relationship between them?
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
Use the information on these cards to draw the shape that is being described.
Can you draw a square in which the perimeter is numerically equal
to the area?
What fractions of the largest circle are the two shaded regions?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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 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.
What do these two triangles have in common? How are they related?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
This article for teachers gives some food for thought when teaching
ideas about area.
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?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Determine the total shaded area of the 'kissing triangles'.
These practical challenges are all about making a 'tray' and covering it with paper.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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
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?
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. . . .
The area of a square inscribed in a circle with a unit radius is,
satisfyingly, 2. What is the area of a regular hexagon inscribed in
a circle with a unit radius?
An investigation that gives you the opportunity to make and justify
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?
Derive a formula for finding the area of any kite.
Can you maximise the area available to a grazing goat?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
How many tiles do we need to tile these patios?
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.
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).