Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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 do these two triangles have in common? How are they related?
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. . . .
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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. . . .
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.
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 practical challenges are all about making a 'tray' and covering it with paper.
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
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?
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?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .
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 help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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. . . .
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Can you draw a square in which the perimeter is numerically equal
to the area?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
An investigation that gives you the opportunity to make and justify
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?
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?
A follow-up activity to Tiles in the Garden.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
How many tiles do we need to tile these patios?
Can you work out the area of the inner square and give an
explanation of how you did it?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
This article for teachers gives some food for thought when teaching
ideas about area.
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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
A simple visual exploration into halving and doubling.
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.