mei conference 2017mei.org.uk/files/conference17/session-l6.pdf · quadrature of the circle...

MEI Conference 2017

Squaring the Circle

and Other Shapes

Kevin Lord kevin.lord@mei.org.uk

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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