Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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 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.
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
These practical challenges are all about making a 'tray' and covering it with paper.
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.
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 do these two triangles have in common? How are they related?
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
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. . . .
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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 is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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. . . .
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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
Bluey-green, white and transparent squares with a few odd bits of
shapes around the perimeter. But, how many squares are there of
each type in the complete circle? Study the picture and make. . . .
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Can you maximise the area available to a grazing goat?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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?
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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Can you draw a square in which the perimeter is numerically equal
to the area?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
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.
A task which depends on members of the group noticing the needs of
others and responding.
What happens to the area and volume of 2D and 3D shapes when you
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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.
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?
I cut this square into two different shapes. What can you say about
the relationship between them?
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.