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?
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?
Explore one of these five pictures.
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
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
A follow-up activity to Tiles in the Garden.
This article for teachers gives some food for thought when teaching
ideas about area.
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.
Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!
How would you move the bands on the pegboard to alter these shapes?
Can you draw a square in which the perimeter is numerically equal
to the area?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
Here are many ideas for you to investigate - all linked with the
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
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
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?
What happens to the area and volume of 2D and 3D shapes when you
A task which depends on members of the group noticing the needs of
others and responding.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
An investigation that gives you the opportunity to make and justify
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.
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.
Can you work out the area of the inner square and give an
explanation of how you did it?
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?
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).
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. . . .
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Derive a formula for finding the area of any kite.
Determine the total shaded area of the 'kissing triangles'.
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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
What fractions of the largest circle are the two shaded regions?
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
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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. . . .
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.
Bluey-green, white and transparent squares with a few odd bits of
shapes around the perimeter. But, how many squares are there of
each type in the complete circle? Study the picture and make. . . .