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.
This article for teachers gives some food for thought when teaching
ideas about area.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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?
A simple visual exploration into halving and doubling.
How would you move the bands on the pegboard to alter these shapes?
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.
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. . . .
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.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.
Can you draw a square in which the perimeter is numerically equal
to the area?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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
Can you maximise the area available to a grazing goat?
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?
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?
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?
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 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?
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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 do these two triangles have in common? How are they related?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you work out the area of the inner square and give an
explanation of how you did it?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in
the middle. Cut the carpet into two pieces to make a 10 by 10 foot
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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).
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
I cut this square into two different shapes. What can you say about
the relationship between them?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.