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.
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. . . .
Can you maximise the area available to a grazing goat?
What fractions of the largest circle are the two shaded regions?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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. . . .
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).
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. . . .
If I print this page which shape will require the more yellow ink?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What happens to the area and volume of 2D and 3D shapes when you
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
How efficiently can you pack together disks?
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 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
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
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. . . .
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?
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?
Derive a formula for finding the area of any kite.
Determine the total shaded area of the 'kissing triangles'.