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

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

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?

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

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?

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. . . .

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

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?

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

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.

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

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

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

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?

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

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

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

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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.

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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

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

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?

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

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?

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 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?

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

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.

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?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

A simple visual exploration into halving and doubling.

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

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

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

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

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

Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .

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?

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?

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

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.

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.