pythagoras and geometry

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Book Title : Geometry Mathsfill Author : C.J. Davis Level : GCSE ISBN 9780956014672 GCSE Maths Workbook Applications of Geometry History of Geometry

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Geometry Workbook highlighting applications and history of geometry and mathematics

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Page 1: Pythagoras and Geometry

Book Title : Geometry Mathsfill

Author : C.J. Davis

Level : GCSE

ISBN 9780956014672

GCSE MathsWorkbook

Applicationsof Geometry

History

of Geometry

Page 2: Pythagoras and Geometry

Discoveries of famous mathematiciansRead about :

Euclid ( 365 – 265 BC ) Democritus ( 460 BC – 370 BC )

Page 3: Pythagoras and Geometry

Some of the major contributors in the development of geometry.

Archimedes c. 290 BC

also :

Pythagoras c. 570 BC

Page 4: Pythagoras and Geometry

exactly two-thirds that of the cylinder surrounding it.

the volume and surface area of a sphere is

proved that . . .

Archimedes

Page 5: Pythagoras and Geometry

Archimedes was the first person

to give the exact expression for

the volume of a sphere . . .

43 r

3 V =

Page 6: Pythagoras and Geometry

The medal carries a portrait of

Archimedes. The inscription around the

medal is a quote in Latin attributed to Archimedes :

“Rise above oneself and grasp the world.”

The Field’s medal is awarded

to mathematicians of

outstanding achievement.

The Field’s Medal

Page 7: Pythagoras and Geometry

Volume of pyramid, V = x area of base x height

. . . worked on the general formula for the volume of a pyramid :

Democritus

Page 8: Pythagoras and Geometry

For a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

is best known for his Pythagorean theorem which states :

Pythagoras

Page 9: Pythagoras and Geometry

Old Babylonian "hand tablet" illustrating Pythagoras' Theorem and an approximation of the square root of two.

The Babylonians understood this rule

1 000 years earlier, but Pythagoras proved it.

Page 10: Pythagoras and Geometry

. . . created the earliest systems

. of weights and measures ?

WHO . . . ?

. . . devised the Metric system ?

Page 11: Pythagoras and Geometry

The Metric system was created by the French Academy of Science in 1793.

A team of mathematicians and scientists were involved in its creation.

Antoine Lavoisier (1743 -1794)

This included the mathematician Joseph Lagrange and the chemist Antoine Lavoisier.

Page 12: Pythagoras and Geometry

GCSE MathsWorkbook

Applicationsof Geometry

History

of Geometry

Page 13: Pythagoras and Geometry

WHAT . . . ?

. . . is the numerical value . of pi ?

Page 14: Pythagoras and Geometry

HOW . . . ?

is trigonometry used to

calculate the height

of a tree ?

Page 15: Pythagoras and Geometry
Page 16: Pythagoras and Geometry

WHAT . . . ?. . . is the Cornrow Curves Software ?

. . . are the four geometric concepts used to create cornrow hairstyles ?

www.rpi.edu/~eglash/eglash.htm

Page 17: Pythagoras and Geometry

The Cornrow Curves software was created by Prof. Ron

Eglash and allows you use geometrical knowledge from

cornrow hairstyles to create your own simulated cornrow

designs on the computer.

www.rpi.edu/~eglash/eglash.htm

Page 18: Pythagoras and Geometry

What are Fractals ?

Each leaf of this fern is in fact a smaller version of the entire fern. The whole fern is simply made up of reduced versions of itself. 

Fractals are images that exhibit something called self-similarity. 

e.g. something that is made up

of a reduced version of itself.

Page 19: Pythagoras and Geometry

. . . is the link between

fractals, African art,

architecture and designs ?

WHAT . . . ?

Page 20: Pythagoras and Geometry

The brachistochrone curve is an inverted cycloid.

The cycloid occurs as the curve generated by a point on a rolling circle.

HOW . . . ?

. . . is a cycloid generated ?

Page 21: Pythagoras and Geometry

Is the shortest path always the quickest ?

WHICH . . . ?

Set up a straight line path and a brachistochrone and see !

. . . path is the quickest route to the bottom ?

Page 22: Pythagoras and Geometry

  Linear Descent vs. the Brachistochrone

When two balls are dropped from the top of the straight ramp and the

the bottom first, even though the path it travels is longer.

brachistochrone

top of the brachistochrone, the ball on the brachistochrone reaches

Page 23: Pythagoras and Geometry

GCSE MathsWorkbook

Applicationsof Geometry

History

of Geometry

Page 24: Pythagoras and Geometry

- Time saving, write-on format.

- Clearly structured, easy to follow layout.

What are students saying about Geometry Mathsfill ?

Page 25: Pythagoras and Geometry

A skateboard ramp has a height

of 1.05 m and a base of 2.24 m.

Calculate the length of the ramp,

2.24 m

1.05 m

m

…………………………………………….………..…..

……………………………………….… ……………m

…………………………………………….………..…..

Page 26: Pythagoras and Geometry

…………………………………………….……………………...…..

………………………………………………….… ………….……km

…………………………………………….………………….…..…..

From take-off an aeroplane climbs at an angle12.

of 18. When the aeroplane has flown 15 km,

what vertical height has it reached ?

Page 27: Pythagoras and Geometry

Foyles Bookshop Charing Cross Road London WC1 England

Available ( online ) at : www.Foyles.co.uk

Available ( in-store ) at :

ISBN 9780956014672

Geometry Mathsfill

(A5) Notepad format

Author: C.J. Davis

Publisher : Bookfill

Price £ 11.99

Page 28: Pythagoras and Geometry

ISBN 9780956014603

(A5) Notepad format

Algebra Mathsfill

Price £ 11.99

Author: C.J. Davis

Publisher : Bookfill

ALSO available :

log onto www.bookfill.co.uk for more details