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).
Look at the mathematics that is all around us - this circular
window is a wonderful example.
What fractions of the largest circle are the two shaded regions?
An investigation that gives you the opportunity to make and justify
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?
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
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. . . .
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
This article for teachers gives some food for thought when teaching
ideas about area.
How would you move the bands on the pegboard to alter these shapes?
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.
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.
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.
Can you draw a square in which the perimeter is numerically equal
to the area?
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
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
A simple visual exploration into halving and doubling.
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
Use the information on these cards to draw the shape that is being described.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
These practical challenges are all about making a 'tray' and covering it with paper.
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!
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?
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
Can you maximise the area available to a grazing goat?
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 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?
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
Explore one of these five pictures.
A task which depends on members of the group noticing the needs of
others and responding.
A follow-up activity to Tiles in the Garden.
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?