MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle...
Transcript of MEI Conference 2017mei.org.uk/files/conference17/Session-L6.pdf · Quadrature of the circle...
Can you prove that the area of the square and the
rectangle are equal?
Use the triangle HPN to show that area of NPQR = area LMNO
Lunes of Hippocrates
Find the shaded area of the lunes.
Geometrical Calculations
What is the relationship between a and x?
Can you construct this to verify your solution?
Squaring the
Circle and
Other Shapes
Kevin Lord
The Greek geometers were more interested in finding a
unifying system for finding the area of any plane shape.
Area was considered as a property of the shape. A square
figure, the most fundamental shape, was both equal to its
area and could be defined by its area.
Greek GeometryBefore the Ancient Greeks, Babylonian and
Egyptian mathematicians were able to
calculate the areas of various plane shapes.
These calculations had practical applications
in working out land usage etc. and required
measurement.
Quadrature Quadrature (or squaring) of a plane shape is the
constructions – using only compass and straight-
edge – of a square with the same area as the
original figure.
Compass and Straight-Edge
• Perpendicular line
• Dropping a perpendicular from a point
• Bisecting a line
• Bisecting an angle
• Marking equidistant points
• Calculating
Quadrature of a rectangle• Construct (or draw ) an “arbitrary”,
say 9cm x 4cm, rectangle - labelled LMNO
• Extend the line MN
• Use a compass to mark off segment NG equal in
length to ON
• Find midpoint MG – point H
• Using H as the centre, draw an arc through M
and G.
• Extend line ON to intersect the arc at P
• NP is one side of the square
Can you prove that the areas are equal?
Use the triangle HPN to show that
area of NPQR = area LMNO
ProofLet a, b, c be the lengths of triangle
HP, HN and PN.
Pythagoras theorem a2 = b2 + c2
a
b
c
Now NG = ON = a - b and MN = a + b.
Area (rectangle LMNO) = MN x ON = (a + b)(a - b)
= a2 - b2
= c2 = Area (square NPQR)
a - b
Quadrature of a triangle
How could you use the method for a rectangle to
construct the square of equal area to the triangle?
Describe the steps
A B
C
Quadrature of a triangle
• Construct (or draw) an arbitrary triangle, ABC
• Drop a perpendicular line from C to the base
• Find the midpoint of CD
A B
C
D
Quadrature of a triangle
• Construct perpendicular through midpoint of CD
• Construct perpendiculars to base through A and
B to complete the rectangle
A B
C
D
Quadrature of a curved shapeOne of the famous problems from antiquity was
how to construct a square with the same area as a
circle.
“squaring the circle”
Squaring the Circle
Hippocrates of Chios c.470-410 BCE
Hippocrates’ investigated the
quadrature of lunes.
Lune
a plane shape
bounded by two
circular arcs.
Lunes of HippocratesFind the shaded area of the lunes.
10
8
Quadrature of a lune
In general, show that the area of the lunes is
equal to the area of triangle ABC.
Lunes of Hippocrates
= =+
+ =
Proof by pictures
Lunes and the Regular Hexagon If a regular hexagon is inscribed in a circle and six
semicircles constructed on its sides, then the area
of the hexagon equals the area of the six lunes
plus the area of a circle whose diameter is equal
in length to one of the sides of the hexagon.
Hippocrates of Chios, ca. 440 B.C.E
Proof by
pictures
Quadrature of the circle
Hippocrates’ work with lunes offered some hope
that there may be a generalisation of his method
leading to squaring the circle.
In 1882, the German mathematician Ferdinand
Lindemann proved that the quadrature of the circle
is impossible by proving that 𝜋 is a
“transcendental number.”
Indiana Pi Bill
Indiana 1897
Edwin Goodwin proposed a bill to the State
Assembly which included a solution to the
problem of squaring the circle.
The bill would have defined 𝝅 = 𝟑. 𝟐 in Indiana.
It was eventually thrown out.
Geometrical Calculations• What is the relationship between a and x?
• Can you construct this to verify your solution?
Geometrical Calculations
p q
Geometrical CalculationsConstruction
• Draw horizontal line across lower part of the
page
• Mark points A, B and C (AB= 12cm, BC = 3cm)
• Construct perpendicular line through B
• Construct bisector for AC (Mark it O)
• Draw a semi-circle, centre O, radius AO
• Measure X
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