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?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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
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. . . .
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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?
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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.
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.
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
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.
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
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
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.
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!
Can you draw the height-time chart as this complicated vessel fills
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
A task which depends on members of the group noticing the needs of
others and responding.
What is the same and what is different about these circle
questions? What connections can you make?
How efficiently can you pack together disks?
Analyse these beautiful biological images and attempt to rank them in size order.
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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. . . .
What happens to the area and volume of 2D and 3D shapes when you
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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. . . .
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.
What fractions of the largest circle are the two shaded regions?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Can you work out the area of the inner square and give an
explanation of how you did it?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?