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.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Measure problems for inquiring primary learners.
Measure problems for primary learners to work on with others.
Measure problems at primary level that require careful consideration.
Measure problems at primary level that may require determination.
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?
What happens to the area and volume of 2D and 3D shapes when you
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.
I cut this square into two different shapes. What can you say about
the relationship between them?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
How would you move the bands on the pegboard to alter these shapes?
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.
This article for teachers gives some food for thought when teaching
ideas about area.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
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?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A simple visual exploration into halving and doubling.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
Use the information on these cards to draw the shape that is being described.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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).
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?
These practical challenges are all about making a 'tray' and covering it with paper.
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!
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?
Are these statements always true, sometimes true or never true?
Can you maximise the area available to a grazing goat?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
An investigation that gives you the opportunity to make and justify
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.
A task which depends on members of the group noticing the needs of
others and responding.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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?