Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Can you draw a square in which the perimeter is numerically equal to the area?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
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?
An investigation that gives you the opportunity to make and justify predictions.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
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?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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 smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
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?
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 the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
I cut this square into two different shapes. What can you say about the relationship between them?
Here are many ideas for you to investigate - all linked with the number 2000.
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?
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?
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
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 centimetres of rope will I need to make another mat just like the one I have here?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
This article for teachers gives some food for thought when teaching ideas about 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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
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?
Can you work out the area of the inner square and give an explanation of how you did it?
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 happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Measure problems at primary level that require careful consideration.
Measure problems for primary learners to work on with others.
Measure problems for inquiring primary learners.
Measure problems at primary level that may require resilience.
Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?
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?
What do these two triangles have in common? How are they related?
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.
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?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
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. . . .