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

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

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

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?

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

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.

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

A follow-up activity to Tiles in the Garden.

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.

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?

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.

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

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

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!

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?

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

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

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

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

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

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

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?

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.

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

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).

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. . . .

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?

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. . . .

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

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

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.

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

Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .

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.

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

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?

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

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

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.

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 simple visual exploration into halving and doubling.

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

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?