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.
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.
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.
Use the information on these cards to draw the shape that is being described.
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.
How would you move the bands on the pegboard to alter these shapes?
A simple visual exploration into halving and doubling.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Can you draw a square in which the perimeter is numerically equal
to the area?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
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
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?
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?
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?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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.
These practical challenges are all about making a 'tray' and covering it with paper.
What do these two triangles have in common? How are they related?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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 shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
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?
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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Derive a formula for finding the area of any kite.
These pictures were made by starting with a square, finding the
half-way point on each side and joining those points up. You could
investigate your own starting shape.
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Explore one of these five pictures.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?
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. . . .
Can you maximise the area available to a grazing goat?
Can you work out the area of the inner square and give an
explanation of how you did it?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.