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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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).
What fractions of the largest circle are the two shaded regions?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Can you maximise the area available to a grazing goat?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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. . . .
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
If I print this page which shape will require the more yellow ink?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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. . . .
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the 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
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?
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!
A task which depends on members of the group noticing the needs of
others and responding.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
What happens to the area and volume of 2D and 3D shapes when you
Explore one of these five pictures.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
A follow-up activity to Tiles in the Garden.
What is the same and what is different about these circle
questions? What connections can you make?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you find the area of a parallelogram defined by two vectors?
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?
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?
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.
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
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?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
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. . . .