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.
How efficiently can you pack together disks?
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?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
A follow-up activity to Tiles in the Garden.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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!
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.
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
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?
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.
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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. . . .
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
What happens to the area and volume of 2D and 3D shapes when you
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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 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.
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.
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?
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
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
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?
Can you maximise the area available to a grazing goat?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
What is the same and what is different about these circle
questions? What connections can you make?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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 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
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.
Can you find the area of a parallelogram defined by two vectors?
A task which depends on members of the group noticing the needs of
others and responding.
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 the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
Draw two circles, each of radius 1 unit, so that each circle goes
through the centre of the other one. What is the area of the
Can you work out the area of the inner square and give an
explanation of how you did it?
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. . . .