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
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
How many tiles do we need to tile these patios?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Can you draw a square in which the perimeter is numerically equal
to the area?
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
If I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?
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?
How would you move the bands on the pegboard to alter these shapes?
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.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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 shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
This article for teachers gives some food for thought when teaching
ideas about area.
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.
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
Here are many ideas for you to investigate - all linked with the
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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. . . .
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?
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
Use the information on these cards to draw the shape that is being described.
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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.
A simple visual exploration into halving and doubling.
What fractions of the largest circle are the two shaded regions?
I cut this square into two different shapes. What can you say about
the relationship between them?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
Can you maximise the area available to a grazing goat?
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?