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?

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).

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

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?

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?

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?

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

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

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

I cut this square into two different shapes. What can you say about the relationship between them?

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

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?

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

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

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.

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.

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?

Measure problems at primary level that may require determination.

Here are many ideas for you to investigate - all linked with the number 2000.

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

Can you draw a square in which the perimeter is numerically equal to the area?

Measure problems at primary level that require careful consideration.

Measure problems for inquiring primary learners.

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!

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

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?

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

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.

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

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 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.

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.

Measure problems for primary learners to work on with others.

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. . . .

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?

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

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.

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

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. . . .

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?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Are these statements always true, sometimes true or never true?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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. . . .

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

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