These practical challenges are all about making a 'tray' and covering it with paper.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
How many tiles do we need to tile these patios?
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
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 fractions of the largest circle are the two shaded regions?
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 practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
Can you draw a square in which the perimeter is numerically equal
to the area?
Here are many ideas for you to investigate - all linked with the
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.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
An investigation that gives you the opportunity to make and justify
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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?
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.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
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 is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
I cut this square into two different shapes. What can you say about
the relationship between them?
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
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. . . .
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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. . . .
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?