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

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

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

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

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

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

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?

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

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.

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?

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

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

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

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?

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

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

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

A simple visual exploration into halving and doubling.

A follow-up activity to Tiles in the Garden.

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?

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

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

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.

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

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?

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

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

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

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

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

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.

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

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.

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.

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

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

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?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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?

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?