Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Can you draw a square in which the perimeter is numerically equal to the area?
An investigation that gives you the opportunity to make and justify predictions.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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?
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?
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?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
I cut this square into two different shapes. What can you say about the relationship between them?
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?
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?
Here are many ideas for you to investigate - all linked with the number 2000.
How many tiles do we need to tile these patios?
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.
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?
Look at the mathematics that is all around us - this circular window is a wonderful example.
A simple visual exploration into halving and doubling.
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.
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?
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How many centimetres of rope will I need to make another mat just like the one I have here?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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 maximise the area available to a grazing goat?
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?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
How would you move the bands on the pegboard to alter these shapes?
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
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?
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?
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.
What do these two triangles have in common? How are they related?
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?
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. . . .
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
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. . . .