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
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?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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?
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. . . .
What do these two triangles have in common? How are they related?
These practical challenges are all about making a 'tray' and covering it with paper.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
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 smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
Can you work out the area of the inner square and give an
explanation of how you did it?
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.
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?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
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. . . .
Can you maximise the area available to a grazing goat?
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?
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. . . .
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 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?
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?
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
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
A simple visual exploration into halving and doubling.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
An investigation that gives you the opportunity to make and justify
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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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!
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
Explore one of these five pictures.