A follow-up activity to Tiles in the Garden.

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.

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.

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

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

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

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?

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

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

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. . . .

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

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?

Determine the total shaded area of the 'kissing triangles'.

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

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

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

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?

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

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

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

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?

Use the information on these cards to draw the shape that is being described.

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.

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!

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

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

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

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

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

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

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?

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. . . .

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).

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?

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.

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

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

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

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

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

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?

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?

A simple visual exploration into halving and doubling.

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.