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.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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!
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.
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
How many tiles do we need to tile these patios?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
A follow-up activity to Tiles in the Garden.
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
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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.
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?
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.
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.
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.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
A simple visual exploration into halving and doubling.
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.
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.
What happens to the area and volume of 2D and 3D shapes when you
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Here are many ideas for you to investigate - all linked with the
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
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.
How would you move the bands on the pegboard to alter these shapes?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
What fractions of the largest circle are the two shaded regions?
Derive a formula for finding the area of any kite.
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).
Can you find rectangles where the value of the area is the same as the value of the perimeter?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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. . . .
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
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?