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
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?
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.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Can you find the area of a parallelogram defined by two vectors?
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. . . .
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.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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?
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.
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 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 maximise the area available to a grazing goat?
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 work out the area of the inner square and give an
explanation of how you did it?
A task which depends on members of the group noticing the needs of
others and responding.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
What happens to the area and volume of 2D and 3D shapes when you
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. . . .
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. . . .
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?
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.
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
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!
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
A follow-up activity to Tiles in the Garden.
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
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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. . . .
Cut off three right angled isosceles triangles to produce a
pentagon. With two lines, cut the pentagon into three parts which
can be rearranged into another square.
Derive a formula for finding the area of any kite.
If I print this page which shape will require the more yellow ink?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?