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?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Determine the total shaded area of the 'kissing triangles'.
How would you move the bands on the pegboard to alter these shapes?
What fractions of the largest circle are the two shaded regions?
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
Use the information on these cards to draw the shape that is being described.
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.
Can you draw a square in which the perimeter is numerically equal
to the area?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
This article for teachers gives some food for thought when teaching
ideas about area.
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
What do these two triangles have in common? How are they related?
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
A follow-up activity to Tiles in the Garden.
A simple visual exploration into halving and doubling.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
These practical challenges are all about making a 'tray' and covering it with paper.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its 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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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).
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. . . .
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
Derive a formula for finding the area of any kite.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
An investigation that gives you the opportunity to make and justify
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
What happens to the area and volume of 2D and 3D shapes when you
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 work out the area of the inner square and give an
explanation of how you did it?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?