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?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
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 squares?
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?
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?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
Explore one of these five pictures.
Look at the mathematics that is all around us - this circular window is a wonderful example.
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 for teachers gives some food for thought when teaching ideas about area.
An investigation that gives you the opportunity to make and justify predictions.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
A task which depends on members of the group noticing the needs of others and responding.
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
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.
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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 time?
Here are many ideas for you to investigate - all linked with the number 2000.
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 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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Derive a formula for finding the area of any kite.
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?
Can you work out the area of the inner square and give an explanation of how you did it?
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?
Determine the total shaded area of the 'kissing triangles'.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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 how many ways can you halve a piece of A4 paper? How do you know they are halves?
Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
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. . . .
Use the information on these cards to draw the shape that is being described.
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 square carpet.
How many tiles do we need to tile these patios?
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 practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
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).
A follow-up activity to Tiles in the Garden.
Measure problems at primary level that require careful consideration.
Measure problems for primary learners to work on with others.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Measure problems for inquiring primary learners.