A follow-up activity to Tiles in the Garden.

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

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.

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?

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. . . .

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?

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

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

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

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?

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

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

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

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

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

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

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

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.

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?

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 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.

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

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

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

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

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?

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

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

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

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

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

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

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

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?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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?

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

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?

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?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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.

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. . . .

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