What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
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?
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.
How many tiles do we need to tile these patios?
A follow-up activity to Tiles in the Garden.
Here are many ideas for you to investigate - all linked with the
What fractions of the largest circle are the two shaded regions?
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
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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
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.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Can you draw a square in which the perimeter is numerically equal
to the area?
Explore one of these five pictures.
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
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
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?
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 task which depends on members of the group noticing the needs of
others and responding.
How many centimetres of rope will I need to make another mat just
like the one I have here?
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.
Determine the total shaded area of the 'kissing triangles'.
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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.
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?
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. . . .
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you work out the area of the inner square and give an
explanation of how you did it?
Derive a formula for finding the area of any kite.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in
the middle. Cut the carpet into two pieces to make a 10 by 10 foot
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).
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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?