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

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

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

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

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. . . .

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).

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

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.

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

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

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

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

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

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?

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?

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

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. . . .

A follow-up activity to Tiles in the Garden.

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?

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

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 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?

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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

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?

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?

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

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.

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

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?

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

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?

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.

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

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?

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

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

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

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

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

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.