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. . . .

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

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

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

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.

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

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

A follow-up activity to Tiles in the Garden.

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

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.

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

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

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

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?

These practical challenges are all about making a 'tray' and covering it with paper.

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

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?

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?

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.

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

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

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.

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?

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

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

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?

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

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?

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

A simple visual exploration into halving and doubling.

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?

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

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.

A task which depends on members of the group noticing the needs of others and responding.

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.

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

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?

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

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

An investigation that gives you the opportunity to make and justify predictions.

This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.

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.

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?

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