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

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

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

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

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.

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?

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

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?

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

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. . . .

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

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

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.

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

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

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?

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?

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

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.

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

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

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. . . .

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

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

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

A follow-up activity to Tiles in the Garden.

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

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 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?

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

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?

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?

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

A simple visual exploration into halving and doubling.

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

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

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

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

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

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.

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