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. . . .
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?
Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Explore one of these five pictures.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
A follow-up activity to Tiles in the Garden.
Can you maximise the area available to a grazing goat?
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?
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
This article for teachers gives some food for thought when teaching ideas about area.
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!
How would you move the bands on the pegboard to alter these shapes?
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.
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.
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
Use the information on these cards to draw the shape that is being described.
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?
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.
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?
What do these two triangles have in common? How are they related?
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?
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
An investigation that gives you the opportunity to make and justify predictions.
Look at the mathematics that is all around us - this circular window is a wonderful example.
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?
How many centimetres of rope will I need to make another mat just like the one I have here?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
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).
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .