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 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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
What fractions of the largest circle are the two shaded regions?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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. . . .
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How would you move the bands on the pegboard to alter these shapes?
What happens to the area and volume of 2D and 3D shapes when you
Can you draw a square in which the perimeter is numerically equal
to the area?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
I cut this square into two different shapes. What can you say about
the relationship between them?
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.
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.
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).
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'.
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 is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Use the information on these cards to draw the shape that is being described.
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?
A simple visual exploration into halving and doubling.
This article for teachers gives some food for thought when teaching
ideas about area.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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!
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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
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?
Can you maximise the area available to a grazing goat?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
A task which depends on members of the group noticing the needs of
others and responding.
How many tiles do we need to tile these patios?
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Are these statements always true, sometimes true or never true?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
An investigation that gives you the opportunity to make and justify
Measure problems at primary level that may require determination.
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 at primary level that require careful consideration.
Measure problems for primary learners to work on with others.