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?
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?
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
These practical challenges are all about making a 'tray' and covering it with paper.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What do these two triangles have in common? How are they related?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
I cut this square into two different shapes. What can you say about
the relationship between them?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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
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. . . .
How many centimetres of rope will I need to make another mat just
like the one I have here?
An investigation that gives you the opportunity to make and justify
Here are many ideas for you to investigate - all linked with the
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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?
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
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?
How many tiles do we need to tile these patios?
Can you maximise the area available to a grazing goat?
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.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
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.
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?
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
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. . . .
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?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
I'm thinking of a rectangle with an area of 24. What could its perimeter be?