My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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.
What do these two triangles have in common? How are they related?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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.
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Can you maximise the area available to a grazing goat?
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. . . .
Look at the mathematics that is all around us - this circular
window is a wonderful example.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 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?
Can you draw a square in which the perimeter is numerically equal
to the area?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
This article for teachers gives some food for thought when teaching
ideas about area.
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.
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?
An investigation that gives you the opportunity to make and justify
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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
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?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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 practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How would you move the bands on the pegboard to alter these shapes?
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 can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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!
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Explore one of these five pictures.
How many tiles do we need to tile these patios?
A follow-up activity to Tiles in the Garden.
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?
What happens to the area and volume of 2D and 3D shapes when you
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you find the area of a parallelogram defined by two vectors?