In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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
How efficiently can you pack together disks?
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
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?
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
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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.
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
Can you maximise the area available to a grazing goat?
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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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. . . .
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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. . . .
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
A point P is selected anywhere inside an equilateral triangle. What
can you say about the sum of the perpendicular distances from P to
the sides of the triangle? Can you prove your conjecture?
Can you find the area of a parallelogram defined by two vectors?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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.
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.
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.
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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?
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. . . .
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
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. . . .
Determine the total shaded area of the 'kissing triangles'.
If I print this page which shape will require the more yellow ink?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
Derive a formula for finding the area of any kite.
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
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.