Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
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. . . .
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
A trapezium is divided into four triangles by its diagonals.
Suppose the two triangles containing the parallel sides have areas
a and b, what is the area of the trapezium?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
If I print this page which shape will require the more yellow ink?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.
Analyse these beautiful biological images and attempt to rank them in size order.
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?
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
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
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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 draw the height-time chart as this complicated vessel fills
Can you find rectangles where the value of the area is the same as the value of the perimeter?
How efficiently can you pack together disks?
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. . . .
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Three squares are drawn on the sides of a triangle ABC. Their areas
are respectively 18 000, 20 000 and 26 000 square centimetres. If
the outer vertices of the squares are joined, three more. . . .
A follow-up activity to Tiles in the Garden.
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?
What happens to the area and volume of 2D and 3D shapes when you
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
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Can you find the area of a parallelogram defined by two vectors?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
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.
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.
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 is the same and what is different about these circle
questions? What connections can you make?
A task which depends on members of the group noticing the needs of
others and responding.
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. . . .
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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.
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 maximise the area available to a grazing goat?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?