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 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.
How many tiles do we need to tile these patios?
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
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
A follow-up activity to Tiles in the Garden.
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!
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
An investigation that gives you the opportunity to make and justify
What happens to the area and volume of 2D and 3D shapes when you
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
A task which depends on members of the group noticing the needs of
others and responding.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Here are many ideas for you to investigate - all linked with the
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 draw a square in which the perimeter is numerically equal
to the area?
How would you move the bands on the pegboard to alter these shapes?
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?
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.
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?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
This article for teachers gives some food for thought when teaching
ideas about area.
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Derive a formula for finding the area of any kite.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
Determine the total shaded area of the 'kissing triangles'.