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
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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.
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. . . .
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you find the area of a parallelogram defined by two vectors?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you maximise the area available to a grazing goat?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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).
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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What is the same and what is different about these circle
questions? What connections can you make?
Determine the total shaded area of the 'kissing triangles'.
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.
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?
Explore one of these five pictures.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you draw the height-time chart as this complicated vessel fills
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
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?
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?
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!
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?
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?
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
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Can you work out the area of the inner square and give an
explanation of how you did 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
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
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.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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. . . .