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. . . .
A simple visual exploration into halving and doubling.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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
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.
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?
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
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?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Can you maximise the area available to a grazing goat?
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.
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
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.
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Explore one of these five pictures.
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?
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?
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.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
What do these two triangles have in common? How are they related?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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).
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 fractions of the largest circle are the two shaded regions?
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
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. . . .
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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
Derive a formula for finding the area of any kite.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?