Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Can you draw the height-time chart as this complicated vessel fills
Analyse these beautiful biological images and attempt to rank them in size order.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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!
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?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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. . . .
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.
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. . . .
Can you maximise the area available to a grazing goat?
If I print this page which shape will require the more yellow ink?
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
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
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
How efficiently can you pack together disks?
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
Six circular discs are packed in different-shaped boxes so that the
discs touch their neighbours and the sides of the box. Can you put
the boxes in order according to the areas of their bases?
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?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
What is the same and what is different about these circle
questions? What connections can you make?
Take any rectangle ABCD such that AB > BC. The point P is on AB
and Q is on CD. Show that there is exactly one position of P and Q
such that APCQ is a rhombus.
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?
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 properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Can you find the area of a parallelogram defined by two vectors?
What happens to the area and volume of 2D and 3D shapes when you
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.
A task which depends on members of the group noticing the needs of
others and responding.
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.
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
A farmer has a field which is the shape of a trapezium as
illustrated below. To increase his profits he wishes to grow two
different crops. To do this he would like to divide the field into
two. . . .
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?