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.
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?
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?
This article for teachers gives some food for thought when teaching
ideas about area.
A simple visual exploration into halving and doubling.
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 would you move the bands on the pegboard to alter these shapes?
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.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Can you draw a square in which the perimeter is numerically equal
to the area?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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 ways can you find of tiling the square patio, using square
tiles of different sizes?
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. . . .
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?
A follow-up activity to Tiles in the Garden.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Can you work out the area of the inner square and give an
explanation of how you did it?
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
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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 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 shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
What do these two triangles have in common? How are they related?
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.
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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!
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
How many tiles do we need to tile these patios?
Explore one of these five pictures.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
These practical challenges are all about making a 'tray' and covering it with paper.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
I cut this square into two different shapes. What can you say about
the relationship between them?