An investigation that gives you the opportunity to make and justify
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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. . . .
Can you draw a square in which the perimeter is numerically equal
to the area?
What fractions of the largest circle are the two shaded regions?
How many tiles do we need to tile these patios?
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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
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?
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.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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.
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.
A tower of squares is built inside a right angled isosceles
triangle. The largest square stands on the hypotenuse. What
fraction of the area of the triangle is covered by the series of
How would you move the bands on the pegboard to alter these shapes?
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
Measure problems for primary learners to work on with others.
A simple visual exploration into halving and doubling.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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.
Are these statements always true, sometimes true or never true?
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?
I cut this square into two different shapes. What can you say about
the relationship between them?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Measure problems at primary level that may require determination.
Measure problems at primary level that require careful consideration.
Measure problems for inquiring primary learners.
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?
How many centimetres of rope will I need to make another mat just
like the one I have here?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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. . . .
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.
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. . . .
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?