A task which depends on members of the group noticing the needs of
others and responding.
What happens to the area and volume of 2D and 3D shapes when you
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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 simple visual exploration into halving and doubling.
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. . . .
Can you maximise the area available to a grazing goat?
How would you move the bands on the pegboard to alter these shapes?
This article for teachers gives some food for thought when teaching
ideas about 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.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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 do these two triangles have in common? How are they related?
Use the information on these cards to draw the shape that is being described.
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
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?
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?
Are these statements always true, sometimes true or never true?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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
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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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).
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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. . . .
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
What fractions of the largest circle are the two shaded regions?
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. . . .