Analyse these beautiful biological images and attempt to rank them in size order.
How efficiently can you pack together disks?
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
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
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. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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
A point P is selected anywhere inside an equilateral triangle. What
can you say about the sum of the perpendicular distances from P to
the sides of the triangle? Can you prove your conjecture?
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?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
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. . . .
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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.
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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
Given a square ABCD of sides 10 cm, and using the corners as
centres, construct four quadrants with radius 10 cm each inside the
square. The four arcs intersect at P, Q, R and S. Find the. . . .
Derive a formula for finding the area of any kite.
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. . . .
What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A task which depends on members of the group noticing the needs of
others and responding.
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 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
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?
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.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
What is the same and what is different about these circle
questions? What connections can you make?
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.
Can you find the area of a parallelogram defined by two vectors?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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?
Can you maximise the area available to a grazing goat?
What happens to the area and volume of 2D and 3D shapes when you
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?
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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!