Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
How many centimetres of rope will I need to make another mat just like the one I have here?
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.
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.
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 measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
How would you move the bands on the pegboard to alter these shapes?
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
How many tiles do we need to tile these patios?
A simple visual exploration into halving and doubling.
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.
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its 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.
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 time?
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?
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 is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
Can you draw a square in which the perimeter is numerically equal to the area?
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
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?
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 into?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
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 this area?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
An investigation that gives you the opportunity to make and justify predictions.
Can you work out the area of the inner square and give an explanation of how you did it?
Here are many ideas for you to investigate - all linked with the number 2000.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Explore one of these five pictures.
Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
A follow-up activity to Tiles in the Garden.
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 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?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Derive a formula for finding the area of any kite.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.