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 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?

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?

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

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!

A follow-up activity to Tiles in the Garden.

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.

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

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?

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

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

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

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

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

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

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.

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

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

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

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?

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?

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. . . .

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

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

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.

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

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. . . .

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

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

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.

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

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. . . .

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

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?

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?

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

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.

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

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

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

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