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?
I cut this square into two different shapes. What can you say about
the relationship between them?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
Can you draw a square in which the perimeter is numerically equal
to the area?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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 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?
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.
Can you maximise the area available to a grazing goat?
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?
How would you move the bands on the pegboard to alter these shapes?
Here are many ideas for you to investigate - all linked with the
This article for teachers gives some food for thought when teaching
ideas about area.
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?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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.
How many tiles do we need to tile these patios?
A simple visual exploration into halving and doubling.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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.
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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Derive a formula for finding the area of any kite.
Determine the total shaded area of the 'kissing triangles'.
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios 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. . . .
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
A follow-up activity to Tiles in the Garden.
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in
the middle. Cut the carpet into two pieces to make a 10 by 10 foot