Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
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?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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?
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. . . .
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
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!
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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 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?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Derive a formula for finding the area of any kite.
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?
Analyse these beautiful biological images and attempt to rank them in size order.
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
Explore one of these five pictures.
Can you draw the height-time chart as this complicated vessel fills
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
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?
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.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
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. . . .
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).
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you maximise the area available to a grazing goat?
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?
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
Can you find the area of a parallelogram defined by two vectors?
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
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?
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.
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
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 has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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?
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?
If I print this page which shape will require the more yellow ink?
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
What is the same and what is different about these circle
questions? What connections can you make?