Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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 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. . . .
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?
If I print this page which shape will require the more yellow ink?
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?
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 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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Can you draw the height-time chart as this complicated vessel fills
A follow-up activity to Tiles in the Garden.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Analyse these beautiful biological images and attempt to rank them in size order.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
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. . . .
Explore one of these five pictures.
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?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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).
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
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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.
Three squares are drawn on the sides of a triangle ABC. Their areas
are respectively 18 000, 20 000 and 26 000 square centimetres. If
the outer vertices of the squares are joined, three more. . . .
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. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Determine the total shaded area of the 'kissing triangles'.
Can you maximise the area available to a grazing goat?
Can you find the area of a parallelogram defined by two vectors?
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!
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?
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.
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.
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 work out the area of the inner square and give an
explanation of how you did it?
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?