What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

Can you draw a square in which the perimeter is numerically equal to the area?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

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.

What is the smallest number of tiles needed to tile this patio? Can you investigate patios 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?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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.

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?

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 centimetres of rope will I need to make another mat just like the one I have here?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

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?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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.

I cut this square into two different shapes. What can you say about the relationship between them?

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.

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?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

Here are many ideas for you to investigate - all linked with the number 2000.

What fractions of the largest circle are the two shaded regions?

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

This article for teachers gives some food for thought when teaching ideas about 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 this 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.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Measure problems for primary learners to work on with others.

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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.

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

A simple visual exploration into halving and doubling.

Look at the mathematics that is all around us - this circular window is a wonderful example.

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?

What do these two triangles have in common? How are they related?

Measure problems at primary level that may require determination.