A task which depends on members of the group noticing the needs of
others and responding.
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.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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. . . .
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. . . .
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?
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.
Can you maximise the area available to a grazing goat?
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. . . .
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
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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.
A simple visual exploration into halving and doubling.
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 happens to the area and volume of 2D and 3D shapes when you
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?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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.
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!
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
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?
Explore one of these five pictures.
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?
A follow-up activity to Tiles in the Garden.
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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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 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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
How would you move the bands on the pegboard to alter these shapes?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in
the middle. Cut the carpet into two pieces to make a 10 by 10 foot
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Can you work out the area of the inner square and give an
explanation of how you did it?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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. . . .