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?

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?

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.

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. . . .

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

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

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?

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

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?

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

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

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

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

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

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.

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

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.

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?

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

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

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

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. . . .

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?

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.

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

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

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?

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

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

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?

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

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. . . .

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. . . .

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

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?

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

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.