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
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?
How many tiles do we need to tile these patios?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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.
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?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Can you draw a square in which the perimeter is numerically equal
to the area?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
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 is the smallest number of tiles needed to tile this patio? Can
you investigate patios 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
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?
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.
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
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.
Measure problems for primary learners to work on with others.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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 simple visual exploration into halving and doubling.
Use the information on these cards to draw the shape that is being described.
These practical challenges are all about making a 'tray' and covering it with paper.
I cut this square into two different shapes. What can you say about
the relationship between them?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Measure problems at primary level that may require determination.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Measure problems at primary level that require careful consideration.
Here are many ideas for you to investigate - all linked with the
Measure problems for inquiring primary learners.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
How would you move the bands on the pegboard to alter these shapes?
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?
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?
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
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.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What fractions of the largest circle are the two shaded regions?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.