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

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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?

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

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

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?

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

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

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

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.

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

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

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

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.

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

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

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?

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

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?

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. . . .

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.

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

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

A follow-up activity to Tiles in the Garden.

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?

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

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

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

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

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. . . .

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

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

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

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?

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

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

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?

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?

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

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.

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?