An investigation that gives you the opportunity to make and justify
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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?
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
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Here are many ideas for you to investigate - all linked with the
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.
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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
This article for teachers gives some food for thought when teaching
ideas about area.
Can you maximise the area available to a grazing goat?
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. . . .
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?
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
How many tiles do we need to tile these patios?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
A simple visual exploration into halving and doubling.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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 fractions of the largest circle are the two shaded regions?
I cut this square into two different shapes. What can you say about
the relationship between them?
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?
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 find the area of a parallelogram defined by two vectors?
What do these two triangles have in common? How are they related?
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?
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.
Explore one of these five pictures.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
How would you move the bands on the pegboard to alter these shapes?