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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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.
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.
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.
I cut this square into two different shapes. What can you say about
the relationship between them?
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
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. . . .
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
Explore one of these five pictures.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
A follow-up activity to Tiles in the Garden.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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
Can you work out the area of the inner square and give an
explanation of how you did it?
Determine the total shaded area of the 'kissing triangles'.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
Derive a formula for finding the area of any kite.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
A task which depends on members of the group noticing the needs of
others and responding.
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!
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.
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?
What happens to the area and volume of 2D and 3D shapes when you
What do these two triangles have in common? How are they related?
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Can you maximise the area available to a grazing goat?
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
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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?