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?
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?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Explore one of these five pictures.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
An investigation that gives you the opportunity to make and justify
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?
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
Look at the mathematics that is all around us - this circular
window is a wonderful example.
A tower of squares is built inside a right angled isosceles
triangle. The largest square stands on the hypotenuse. What
fraction of the area of the triangle is covered by the series of
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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
Can you draw a square in which the perimeter is numerically equal
to the area?
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.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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!
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?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
This article for teachers gives some food for thought when teaching
ideas about area.
Derive a formula for finding the area of any kite.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
A simple visual exploration into halving and doubling.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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. . . .
What do these two triangles have in common? How are they related?
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?
What happens to the area and volume of 2D and 3D shapes when you
A task which depends on members of the group noticing the needs of
others and responding.
I cut this square into two different shapes. What can you say about
the relationship between them?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?