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?
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.
What do these two triangles have in common? How are they related?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
Determine the total shaded area of the 'kissing triangles'.
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 practical challenges are all about making a 'tray' and covering it with paper.
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?
I cut this square into two different shapes. What can you say about
the relationship between them?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
How many ways can you find of tiling the square patio, using square
tiles 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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Use the information on these cards to draw the shape that is being described.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
Derive a formula for finding the area of any kite.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How would you move the bands on the pegboard to alter these shapes?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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?
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.
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 maximise the area available to a grazing goat?
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?
This article for teachers gives some food for thought when teaching
ideas about area.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Explore one of these five pictures.
How many tiles do we need to tile these patios?
A follow-up activity to Tiles in the Garden.
A simple visual exploration into halving and doubling.
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
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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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. . . .
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. . . .
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.