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
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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?
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. . . .
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
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
What do these two triangles have in common? How are they related?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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. . . .
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.
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
How many centimetres of rope will I need to make another mat just
like the one I have here?
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. . . .
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Can you maximise the area available to a grazing goat?
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?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
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?
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?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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
Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.
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?
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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
An investigation that gives you the opportunity to make and justify
This article for teachers gives some food for thought when teaching
ideas about area.
A simple visual exploration into halving and doubling.
How would you move the bands on the pegboard to alter these shapes?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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. . . .
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. . . .
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?