Can you draw a square in which the perimeter is numerically equal to the area?

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

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.

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

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

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?

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

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?

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?

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

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

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

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

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

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.

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?

Here are many ideas for you to investigate - all linked with the number 2000.

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

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

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

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?

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.

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 the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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?

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

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?

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

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.

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

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

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?

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

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. . . .

A simple visual exploration into halving and doubling.

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

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

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

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. . . .

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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?

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?

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