Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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.
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. . . .
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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.
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
If I print this page which shape will require the more yellow ink?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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.
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
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 show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
Can you maximise the area available to a grazing goat?
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. . . .
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.
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.
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?
What is the same and what is different about these circle
questions? What connections can you make?
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
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
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?
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.
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. . . .
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?
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.
Determine the total shaded area of the 'kissing triangles'.
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What happens to the area and volume of 2D and 3D shapes when you
Cut off three right angled isosceles triangles to produce a
pentagon. With two lines, cut the pentagon into three parts which
can be rearranged into another square.
Can you find the area of a parallelogram defined by two vectors?
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?
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
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
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
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