In how many ways can you halve a piece of A4 paper? How do you know they are halves?

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .

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?

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?

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

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

How would you move the bands on the pegboard to alter these shapes?

Use the information on these cards to draw the shape that is being described.

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?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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 time?

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.

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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?

A simple visual exploration into halving and doubling.

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?

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

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

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

Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?

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

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.

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

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

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

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 smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Determine the total shaded area of the 'kissing triangles'.

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?

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

Can you work out the area of the inner square and give an explanation of how you did it?

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

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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?

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

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

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.

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?

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?

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?

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.