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

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.

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. . . .

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

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?

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

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

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

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?

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?

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

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

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

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.

A follow-up activity to Tiles in the Garden.

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

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

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

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?

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.

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?

Are these statements always true, sometimes true or never true?

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?

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

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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

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

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?

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?

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?

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

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?

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

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?

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

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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.

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. . . .

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.

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

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