It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you work out the area of the inner square and give an
explanation of how you did it?
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
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
Can you maximise the area available to a grazing goat?
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
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
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. . . .
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.
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
A tower of squares is built inside a right angled isosceles
triangle. The largest square stands on the hypotenuse. What
fraction of the area of the triangle is covered by the series of
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Determine the total shaded area of the 'kissing triangles'.
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. . . .
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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.
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.
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?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Can you find rectangles where the value of the area is the same as the value of the 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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
These practical challenges are all about making a 'tray' and covering it with paper.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
This article for teachers gives some food for thought when teaching
ideas about area.
What do these two triangles have in common? How are they related?
A task which depends on members of the group noticing the needs of
others and responding.
An investigation that gives you the opportunity to make and justify
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
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?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Explore one of these five pictures.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
I'm thinking of a rectangle with an area of 24. What could its perimeter be?