How would you move the bands on the pegboard to alter these shapes?
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.
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.
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?
This article for teachers gives some food for thought when teaching
ideas about area.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
A simple visual exploration into halving and doubling.
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.
A follow-up activity to Tiles in the Garden.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Determine the total shaded area of the 'kissing triangles'.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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!
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.
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. . . .
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
Can you work out the area of the inner square and give an
explanation of how you did it?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Explore one of these five pictures.
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).
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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.
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?
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 do these two triangles have in common? How are they related?
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?
How many tiles do we need to tile these patios?
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.
An investigation that gives you the opportunity to make and justify
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
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A task which depends on members of the group noticing the needs of
others and responding.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Can you maximise the area available to a grazing goat?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Derive a formula for finding the area of any kite.
What fractions of the largest circle are the two shaded regions?