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.
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 a right-angled tetrahedron prove that the sum of the squares of
the areas of the 3 faces in mutually perpendicular planes equals
the square of the area of the sloping face. A generalisation. . . .
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.
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
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?
What is the same and what is different about these circle
questions? What connections can you make?
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?
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 I print this page which shape will require the more yellow ink?
A finite area inside and infinite skin! You can paint the interior
of this fractal with a small tin of paint but you could never get
enough paint to paint the edge.
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. . . .
Make a poster using equilateral triangles with sides 27, 9, 3 and 1 units assembled as stage 3 of the Von Koch fractal. Investigate areas & lengths when you repeat a process infinitely often.
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 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
This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
A and B are two points on a circle centre O. Tangents at A and B
cut at C. CO cuts the circle at D. What is the relationship between
areas of ADBO, ABO and ACBO?
Three triangles ABC, CBD and ABD (where D is a point on AC) are all
isosceles. Find all the angles. Prove that the ratio of AB to BC is
equal to the golden ratio.
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.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Find a quadratic formula which generalises Pick's Theorem.
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?
One side of a triangle is divided into segments of length a and b
by the inscribed circle, with radius r. Prove that the area is:
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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.
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?
How efficiently can you pack together disks?
Change the squares in this diagram and spot the property that stays
the same for the triangles. Explain...
Draw three equal line segments in a unit circle to divide the
circle into four parts of equal area.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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.
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?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
Can you draw the height-time chart as this complicated vessel fills
Analyse these beautiful biological images and attempt to rank them in size order.
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?
The square ABCD is split into three triangles by the lines BP and
CP. Find the radii of the three inscribed circles to these
triangles as P moves on AD.
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
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Follow the hints and prove Pick's Theorem.
Can you find the area of a parallelogram defined by two vectors?
Join in this ongoing research. Build squares on the sides of a
triangle, join the outer vertices forming hexagons, build further
rings of squares and quadrilaterals, investigate.