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?
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?
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?
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. . . .
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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?
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.
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
Derive a formula for finding the area of any kite.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
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?
Explore one of these five pictures.
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
How efficiently can you pack together disks?
Can you draw the height-time chart as this complicated vessel fills
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
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?
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.
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).
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
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 maximise the area available to a grazing goat?
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?
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.
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!
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.
Determine the total shaded area of the 'kissing triangles'.
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 rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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.