It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
An investigation that gives you the opportunity to make and justify
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?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
How many centimetres of rope will I need to make another mat just
like the one I have here?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
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?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
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.
Can you maximise the area available to a grazing goat?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
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. . . .
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
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?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
This article for teachers gives some food for thought when teaching
ideas about area.
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.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
I cut this square into two different shapes. What can you say about
the relationship between them?
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.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
How many tiles do we need to tile these patios?
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?
What do these two triangles have in common? How are they related?
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?