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?
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.
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).
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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
What fractions of the largest circle are the two shaded regions?
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?
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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
Can you maximise the area available to a grazing goat?
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. . . .
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?
If I print this page which shape will require the more yellow ink?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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
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?
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.
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
How efficiently can you pack together disks?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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 happens to the area and volume of 2D and 3D shapes when you
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
A task which depends on members of the group noticing the needs of
others and responding.
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!
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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?
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
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. . . .
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. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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
Can you work out the area of the inner square and give an
explanation of how you did it?
Determine the total shaded area of the 'kissing triangles'.