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. . . .

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?

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?

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

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

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 follow-up activity to Tiles in the Garden.

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

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

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

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?

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

Measure problems at primary level that may require determination.

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

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

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

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.

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

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?

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?

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

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?

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.

Measure problems at primary level that require careful consideration.

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

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

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

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?

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

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

A simple visual exploration into halving and doubling.

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

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

Measure problems for inquiring primary learners.

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?

Measure problems for primary learners to work on with others.

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

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.

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?

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

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