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
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. . . .
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
Can you maximise the area available to a grazing goat?
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
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.
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. . . .
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?
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?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
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
How many centimetres of rope will I need to make another mat just
like the one I have here?
Determine the total shaded area of the 'kissing triangles'.
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. . . .
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
A task which depends on members of the group noticing the needs of
others and responding.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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. . . .
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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.
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).
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?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 triangle.
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 do these two triangles have in common? How are they related?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Are these statements always true, sometimes true or never true?
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 practical challenges are all about making a 'tray' and covering it with paper.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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?