What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
An investigation that gives you the opportunity to make and justify
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
Can you draw a square in which the perimeter is numerically equal
to the area?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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?
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?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Measure problems at primary level that may require determination.
Measure problems at primary level that require careful consideration.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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.
This article for teachers gives some food for thought when teaching
ideas about area.
Measure problems for primary learners to work on with others.
Measure problems for inquiring primary learners.
Use the information on these cards to draw the shape that is being described.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Are these statements always true, sometimes true or never true?
These practical challenges are all about making a 'tray' and covering it with paper.
A simple visual exploration into halving and doubling.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
How would you move the bands on the pegboard to alter these shapes?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
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
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?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
What fractions of the largest circle are the two shaded regions?
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