Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Can you draw a square in which the perimeter is numerically equal to the area?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
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?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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?
Here are many ideas for you to investigate - all linked with the number 2000.
An investigation that gives you the opportunity to make and justify predictions.
How many tiles do we need to tile these patios?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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?
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?
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.
Look at the mathematics that is all around us - this circular window is a wonderful example.
What do these two triangles have in common? How are they related?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
How many centimetres of rope will I need to make another mat just like the one I have here?
I cut this square into two different shapes. What can you say about the relationship between them?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
This article for teachers gives some food for thought when teaching ideas about area.
Can you maximise the area available to a grazing goat?
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.
How would you move the bands on the pegboard to alter these shapes?
A follow-up activity to Tiles in the Garden.
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?
Explore one of these five pictures.
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.
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?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
A simple visual exploration into halving and doubling.
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
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.
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you find the area of a parallelogram defined by two vectors?
The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
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?
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?
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. . . .
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.