A task which depends on members of the group noticing the needs of
others and responding.
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?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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. . . .
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
What happens to the area and volume of 2D and 3D shapes when you
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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. . . .
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What do these two triangles have in common? How are they related?
Can you draw a square in which the perimeter is numerically equal
to the area?
Determine the total shaded area of the 'kissing triangles'.
These practical challenges are all about making a 'tray' and covering it with paper.
Use the information on these cards to draw the shape that is being described.
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
Derive a formula for finding the area of any kite.
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 is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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!
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?
This article for teachers gives some food for thought when teaching
ideas about area.
Can you maximise the area available to a grazing goat?
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.
An investigation that gives you the opportunity to make and justify
How many tiles do we need to tile these patios?
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
A simple visual exploration into halving and doubling.
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?
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.
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?