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 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).
Are these statements always true, sometimes true or never true?
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.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Measure problems at primary level that may require determination.
This article for teachers gives some food for thought when teaching
ideas about area.
Measure problems at primary level that require careful consideration.
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.
Can you draw a square in which the perimeter is numerically equal
to the area?
Here are many ideas for you to investigate - all linked with the
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
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.
Measure problems for primary learners to work on with others.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Use the information on these cards to draw the shape that is being described.
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?
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 is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
How many tiles do we need to tile these patios?
An investigation that gives you the opportunity to make and justify
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?
I cut this square into two different shapes. What can you say about
the relationship between them?
What happens to the area and volume of 2D and 3D shapes when you
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Measure problems for inquiring primary learners.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
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. . . .
Bluey-green, white and transparent squares with a few odd bits of
shapes around the perimeter. But, how many squares are there of
each type in the complete circle? Study the picture and make. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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
These practical challenges are all about making a 'tray' and covering it with paper.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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. . . .
What do these two triangles have in common? How are they related?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you work out the area of the inner square and give an
explanation of how you did it?