An investigation that gives you the opportunity to make and justify
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
How many ways can you find of tiling the square patio, using square
tiles 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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Here are many ideas for you to investigate - all linked with the
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
How many tiles do we need to tile these patios?
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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?
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.
Can you maximise the area available to a grazing goat?
This article for teachers gives some food for thought when teaching
ideas about 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.
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?
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
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. . . .
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.
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
What do these two triangles have in common? How are they related?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
Can you work out the area of the inner square and give an
explanation of how you did it?
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
Use the information on these cards to draw the shape that is being described.
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .