A task which depends on members of the group noticing the needs of
others and responding.
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .
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?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
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
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?
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
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. . . .
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
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
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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 happens to the area and volume of 2D and 3D shapes when you
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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.
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
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.
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?
A tower of squares is built inside a right angled isosceles
triangle. The largest square stands on the hypotenuse. What
fraction of the area of the triangle is covered by the series of
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A simple visual exploration into halving and doubling.
These practical challenges are all about making a 'tray' and covering it with paper.
Use the information on these cards to draw the shape that is being described.
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?
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 activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
What do these two triangles have in common? How are they related?
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.
This article for teachers gives some food for thought when teaching
ideas about area.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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).
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
What fractions of the largest circle are the two shaded regions?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Can you work out the area of the inner square and give an
explanation of how you did it?
Determine the total shaded area of the 'kissing triangles'.
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. . . .
Derive a formula for finding the area of any kite.