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

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

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

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

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

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

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.

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?

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

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

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?

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!

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?

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.

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

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

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?

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

A follow-up activity to Tiles in the Garden.

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.

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

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

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

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

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).

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. . . .

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

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?

These practical challenges are all about making a 'tray' and covering it with paper.

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

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?

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?

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

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?

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

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?

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

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

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

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?