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

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 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?

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

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.

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?

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

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?

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

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. . . .

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?

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

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

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

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.

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

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

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. . . .

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

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.

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?

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 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?

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

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

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

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?

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?

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

A simple visual exploration into halving and doubling.

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

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

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. . . .

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

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?

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!

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?

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?

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.