How many centimetres of rope will I need to make another mat just
like the one I have here?
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.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
I cut this square into two different shapes. What can you say about
the relationship between them?
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
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. . . .
An investigation that gives you the opportunity to make and justify
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.
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?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
This article for teachers gives some food for thought when teaching
ideas about area.
What do these two triangles have in common? How are they related?
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?
Can you draw a square in which the perimeter is numerically equal
to the area?
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
How would you move the bands on the pegboard to alter these shapes?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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 at primary level that require careful consideration.
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?
A simple visual exploration into halving and doubling.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
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.
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?
Measure problems for primary learners to work on with others.
Measure problems for inquiring primary learners.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its 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.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
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
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How many tiles do we need to tile these patios?
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. . . .
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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Can you work out the area of the inner square and give an
explanation of how you did it?
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. . . .