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 can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
What fractions of the largest circle are the two shaded regions?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
An investigation that gives you the opportunity to make and justify
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Can you draw a square in which the perimeter is numerically equal
to the area?
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
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
Here are many ideas for you to investigate - all linked with the
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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 is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
How many tiles do we need to tile these patios?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Explore one of these five pictures.
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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?
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 happens to the area and volume of 2D and 3D shapes when you
A task which depends on members of the group noticing the needs of
others and responding.
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.
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.
I cut this square into two different shapes. What can you say about
the relationship between them?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
Can you work out the area of the inner square and give an
explanation of how you did it?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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).
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .