Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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?
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?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
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?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
How many tiles do we need to tile these patios?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
An investigation that gives you the opportunity to make and justify predictions.
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?
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?
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.
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?
Here are many ideas for you to investigate - all linked with the number 2000.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
How would you move the bands on the pegboard to alter these shapes?
This article for teachers gives some food for thought when teaching ideas about area.
Look at the mathematics that is all around us - this circular window is a wonderful example.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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 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?
Can you draw a square in which the perimeter is numerically equal to the area?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
A simple visual exploration into halving and doubling.
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?
I cut this square into two different shapes. What can you say about the relationship between them?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
How many centimetres of rope will I need to make another mat just like the one I have here?
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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 this area?
What do these two triangles have in common? How are they related?
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.
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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 into?
Can you work out the area of the inner square and give an explanation of how you did it?
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. . . .
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?
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.
Can you maximise the area available to a grazing goat?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.