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. . . .

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. . . .

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.

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?

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

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

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

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?

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

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

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?

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

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

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

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?

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.

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!

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.

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

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

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. . . .

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

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).

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

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

A follow-up activity to Tiles in the Garden.

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. . . .

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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

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.

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?

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

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

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?

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.

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

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

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

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

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

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

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

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?

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?

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