Can you find the area of a parallelogram defined by two vectors?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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. . . .
Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).
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?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
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
Determine the total shaded area of the 'kissing triangles'.
What is the same and what is different about these circle
questions? What connections can you make?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Explore one of these five pictures.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
How efficiently can you pack together disks?
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 is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.
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 work out the area of the inner square and give an
explanation of how you did it?
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 follow-up activity to Tiles in the Garden.
Can you maximise the area available to a grazing goat?
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 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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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!
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
A task which depends on members of the group noticing the needs of
others and responding.
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
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
What happens to the area and volume of 2D and 3D shapes when you
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?
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.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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
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
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?