Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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. . . .
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 shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Explore one of these five pictures.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
A follow-up activity to Tiles in the Garden.
What fractions of the largest circle are the two shaded regions?
Can you maximise the area available to a grazing goat?
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.
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
This article for teachers gives some food for thought when teaching
ideas about area.
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!
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Can you find the area of a parallelogram defined by two vectors?
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.
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 do these two triangles have in common? How are they related?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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?
What happens to the area and volume of 2D and 3D shapes when you
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?
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.
A simple visual exploration into halving and doubling.
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?
A task which depends on members of the group noticing the needs of
others and responding.
I cut this square into two different shapes. What can you say about
the relationship between them?
How would you move the bands on the pegboard to alter these shapes?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
An investigation that gives you the opportunity to make and justify
Look at the mathematics that is all around us - this circular
window is a wonderful example.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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
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
Can you work out the area of the inner square and give an
explanation of how you did it?
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?
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 rectangles have been torn. How many squares did each one have
inside it before it was ripped?