Look at the mathematics that is all around us - this circular
window is a wonderful example.
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 article for teachers gives some food for thought when teaching
ideas about area.
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?
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.
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.
A simple visual exploration into halving and doubling.
Use the information on these cards to draw the shape that is being described.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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 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?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
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
Can you draw a square in which the perimeter is numerically equal
to the area?
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.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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. . . .
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?
Explore one of these five pictures.
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
Determine the total shaded area of the 'kissing triangles'.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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 thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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).
A follow-up activity to Tiles in the Garden.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
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 shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
Can you maximise the area available to a grazing goat?
How many tiles do we need to tile these patios?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
I cut this square into two different shapes. What can you say about
the relationship between them?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
An investigation that gives you the opportunity to make and justify
What do these two triangles have in common? How are they related?
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?