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Lesson 10: Solids. Surface Areas. Prisms and Pyramids (II)

May 22, 2012

3. Prisms

Prisms are polyhedrons that have two parallel, equally sized faces called bases and their lateral faces are parallelograms.

prism is a 3D shape which has a constant cross-section – both ends of the solid are the same shape and anywhere you cut parallel to these ends gives you the same shape too.

3.1.Types of prisms

To name a prism we use to refer to the base polygon. This way we can name them by their bases:

Right prisms are those that have lateral faces rectangles or squares, i.e. basic edges are perpendicular to the lateral edges

Oblique prisms are those whose lateral faces are rhomboids.

Oblique prism

Regular prisms are right prism whose bases are regular polygons.

Irregular prisms are prisms whose bases are irregular polygons.

Parallelepipeds are prisms whose bases are parallelograms.

Cuboids are right parallelepipeds, i.e. their faces are rectangular. Cuboids are very common in daily life. They are rectangular prisms.

3.2. Elements of a prism

3.3. Surface area of a prism

The net of a right prism is formed by:

  • A rectangle of length all the sides of the base, this is the perimeter of the base, and width the height of the prism.
  • Two equal polygons, the ones that are the bases.

It is easy to deduce the surface area of a prism from its net. So it will be the sum of the lateral area and the areas of the two equal basis:

  • Lateral area AL: It is the sum of the areas of the lateral faces but, if we look at the net, the lateral surface is a rectangle of length the perimeter of the base PB, and width the height of the prism h.
  • Area of the bases AB: This is the sum of the areas of the bases.

The surface area of a right prism is:

AT = AL + 2·AB =  PB · h + 2·AB

Now, you can see a video about how to find the surface area of a prism.

4. Pyramids

 A pyramid is a polyhedron whose base can be any polygon and whose lateral faces are triangles with a common vertex (apex of the pyramid).

pyramid has sloping sides that meet at a point.

4.1. Elements of a pyramid

4.2. Types of pyramids

To name a pyramid we use to refer to its base polygon. This way we can name them by their bases:



A right pyramid has isosceles triangles as its lateral faces and its apex lies directly above the midpoint of the base.

An oblique pyramid does not have all isosceles triangles as its sides.

Oblique pyramid

A regular pyramid is right pyramid whose base is a regular polygon and its lateral faces are equally sized. In other case is named irregular pyramid.

4.3 Surface area of a regular pyramid

The net of a regular pyramid is formed by:

  • As many isosceles triangles as sides the base has.
  • The base polygon

Let  the number of sides of the base of the regular polygon that is the base of the pyramid.

We deduce the surface area of a prism from its net. So it will be the sum of the lateral area and the area of the base:

  • Lateral area AL: It is the sum of the areas of the lateral faces but, if we look at the net, the lateral face is n isosceles triangles whose base is the side of the polygon, b, of the base and height the apothem of the pyramid. If we add them up, we have:

 

AL = n · (b · a) = (PB · a)/2

 where  PB the perimeter of the base, and a the apothem of the pyramid.

  • Area of the bases AB: As the base is a regular polygon:

 

AB = (PB · a’)/2

where a’ the apothem of the base.

The surface area of a regular pyramid is:

AT = AL + AB =  (PB · a)/2 + PB · a’)/2

Now, you can see a video about how to find the surface area of a pyramid, in this case cuadrangular).

The following links will provide you enough exercises

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Lesson 10: Solids.Surface Areas (I)

May 16, 2012

Vocabulary on Solids 

Lesson 10: Solids. Surface Areas (Notes)

 Practice Problems on Classifying Solids

Practice problems on Surface Area of Solids

I Have to Know by the End of this Lesson

 

Three-D  shapes have 3-dimensions- length, width and depth. We are going to study some of them: polyhedrons and revolution solids.

1. Polyhedrons

A polyhedron is a three-dimensional region of the space bounded by polygons.

Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren’t polyhedrons).

If you click on the links you will learn more about these 3-D shapes in the web where it is taken from the table below, www.mathisfun.com. You can also find questions with answers.

  Polyhedrons :
(they must have flat faces)
tetrahedronhexahedronoctahedrondodecahedornicosahedron Platonic Solids
triangular prismsquare prismpentagonal prism Prisms
triangular pyramidsquare pyramidpentagonal pyramid Pyramids
     
  Non-Polyhedra:
(if any surface is not flat)
sphere Sphere torus Torus
cylinder Cylinder cone Cone

Elements of a polyhedron

Faces: These polygons that limit the polyhedrons.

Edges: Line segments where two faces of a polyhedron meet. They are sides of the faces.

Vertices of the polyhedron: Points at which three or more polyhedron edges of a polyhedron meet.

Diagonals of a polyhedron: Segments, joining two vertices, which are not placed on the same face, are called diagonals of polyhedron. The tetrahedron has no diagonals.

Dihedral angle (also called the face angle) is the internal angle at which two adjacent faces meet. All dihedral angles between the edges are ≤ 180º.

Polyhedral angle: is the portion of space limited by tree or more faces which meet at a vertex. All polyhedral angles between the edges are ≤ 360º .

Net: It is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. There are several possibilities for a net of a polyhedron.

    Tetrahedrom                                                   Net  for a tetrahedron

You can find the net of the main polyhedron and their different possibilities on

http://gwydir.demon.co.uk/jo/solid/index.htm

2. Regular polyhedrons

2.1. Types of polyhedrons

A convex polyhedron is defined as follows: no line segment joining two of its points contains a point belonging to its exterior. There are many examples known by you: the cube, prisms, pyramids….

A concave polyhedron, on the other hand, will have line segments that join two of its points with all but the two points lying in its exterior.

Below is an example of a concave polyhedron.

The study of polyhedrons was a popular study item in Greek geometry even before the time of Plato (427 – 347 B.C.E.) In 1640, Rene Descartes, a French philosopher, mathematician, and scientist, observed the following formula. In 1752, Leonhard Euler, a Swiss mathematician, rediscovered and used it.

In a convex polyhedron:

If  F = number of faces , V = number of vertices and E = number of edges, then

 F + V = E +2

This formula is named Euler’s Formula.

All the convex polyhedrons verify this formula. There are some concave polyhedrons that verify it. However, there are concave polyhedrons that don’t verify this formula.

2.2. Regular polyhedrons

A regular polyhedron is a polyhedron where:

  • each face is the same regular polygon
  • the same number of faces (polygons) meet at each vertex (corner)

They are also called Platonic solids.

There are five Platonic solids:

1. Tetrahedron, which has three equilateral triangles at each corner.
2. Cube, which has three squares at each corner.
3. Octahedron, which has four equilateral triangles at each corner.
4. dodecahedron, which has three regular pentagons at each corner.
5. Icosahedron, which has five equilateral triangles at each corner.

Exercise: Fill in this table and check they satisfy Euler’s formula. Why?

                faces edges vertices
tetrahedron      ___   ___    ___
cube             ___   ___    ___
octahedron       ___   ___    ___
dodecahedron     ___   ___    ___
icosahedron      ___   ___    ___
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History about Ancient Pi Approximations and a Joke

April 25, 2012

A bit of History about Pi

Mainstream historians believe that ancient Egyptians had no concept of π

As early as the 19th century BCE, Babylonian mathematicians were using π ≈ 25/8, which is about 0.5 percent below the exact value

The Indian astronomer Yajnavalkya gave astronomical calculations in the Shatapatha Brahmana (c. 9th century BCE) that led to a fractional approximation of π ≈ 339/108 (which equals 3.13888…, which is correct to two decimal places when rounded, or 0.09 percent below the exact value).

Recall an n-gon is a polygon with n sides.

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach utilizing polygons which was used around 250 BC by Greek mathematician Archimedes.

Archimedes of Syracuse

(Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC)

This polygonal algorithm remained the primary approach for computing π for over 1,000 years. Archimedes computed upper and lower bounds of π by drawing regular polygons inside and outside a circle, and calculating the perimeters of the outer and inner polygons as you can see below.

By using the equivalent of 96-sided polygons, he proved that :

223/71 < π < 22/7.

Archimedes’ upper bound of 22/7 may have led to widespread belief that π was equal to 22/7. Around 150 AD, Greek-Roman scientist Ptolemy, in his book Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.

(From Wikipedia, several articles)

Nowadays we know hundred of thousands of decimal figures of PI, but using computers!


A  joke

Time to relax

See you tomorrow!

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Lesson 8: A Tale about Thales

April 5, 2012

 

Born Approximately 624 BC,Miletus,Asia Minor. (Now Balat,Turkey)
Died Approximately 547 BC

Thales, an engineer by trade, was the first of the Seven Sages, or wise men of Ancient Greece. Thales is known as the first Greek philosopher, mathematician and scientist. He founded the geometry of lines, so is given credit for introducing abstract geometry.

He was the founder of the Ionian school of philosophy in Miletus, and the teacher of Anaximander. During Thales’ time, Miletus was an important Greek metropolis in Asia Minor, known for scholarship. Several schools were founded in Miletus, attracting scientists, philosophers, architects and geographers.

Thales is credited with introducing the concepts of logical proof for abstract propositions.

Thales went to Egypt and studied with the priests, where he learned of mathematical innovations and brought this knowledge back to Greece. Thales also did geometrical research and, using triangles, applied his understanding of geometry to calculate the distance from shore of ships at sea. This was particularly important to the Greeks, whether the ships were coming to trade or to do battle.

But this is too serious, here it is your tale:

About 600 BC Thales was visiting Egypt. The Pyramids were built 2000 years ago. While Thales was in Egypt, he was called by the Pharaoh who knew his fame of wise man. He proposed an old problem:

-“Find out the exact height of the Great Pyramid and I will cover you with gold”

Thales went to the Great Pyramid and leaned on his  stick and waited. When the stick casted a shadow equal to its length he said to a servant:

– “Run quickly and measure the shadow of the Great Pyramid because at this moment it is as long as the Pyramid”.

This way the Pharaoh made him part of Pharaoh court of wise men and covered him with gold.

Thales learned about the Egyptian rope-pullers and their methods of surveying land for the Pharaoh using stakes and ropes. Property boundaries had to be re-established each year after the Nile flooded. After Thales returned to Greece about 585 BC with notes about what he had learned, and Greek mathematicians translated the rope-and-stake methods of the rope pullers into a system of points, lines and arcs. They also took geometry from the fields to the page by employing two drawing tools, the straightedge for straight lines and the compass for arcs

There are many recorded tales about Thales, some complimentary and others critical:

  • Herodotus noted that Thales predicted the solar eclipse of 585 BC, a notable advancement for Greek science.
  • Aristotle reported that Thales used his skills at recognizing weather patterns to predict that the next season’s olive crop would be bountiful. He purchased all the olive presses in the area, and made a fortune when the prediction came true!
  • Plato told a story of Thales gazing at the night sky, not watching where he walked, and so fell into a ditch. The servant girl who came to help him up then said to him “How do you expect to understand what is going on up in the sky if you do not even see what is at your feet?”. Perhaps this is the first absent-minded professor joke in the West!
Quotations attributed to Thales
  • “A multitude of words is no proof of a prudent mind.”
  • “Hope is the poor man’s bread.”
  • “The past is certain, the future obscure.”
  • “Nothing is more active than thought, for it travels over the universe, and nothing is stronger than necessity for all must submit to it.”
  • “Know thyself.”
The historical notes are an abstract from:
 http://www.mathopenref.com/thales.html by Charlene Douglass, California, 2006.
 
You can learn more in that page or in:
 http://www-history.mcs.st-and.ac.uk/Biographies/Thales.html
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Lesson 7: Solution to Diophantus Riddle (VI)

March 17, 2012

This is the translation of the epitaph into Spanish:

” Aquí fueron sepultados los restos de Diofanto. La infancia de Diofanto duró 1/6 de su vida, 1/12 en la adolescencia, cuando la barba cubrió su cara, Después de 1/7 de su vida contrajo nupcias. Luego de cinco años de casado nació su primer hijo. El hijo vivió ½ de la vida de su padre, su padre buscó consuelo en los números pero no lo logró y murió cuatro años después que él.”

Let x be the age of Diophantus. If we translate the epitaph into Algebra:

x/6 +x/12 +x/7 +5 +x/2 +4 = x
14x/84 +7x/84 +12x/84 +420/84 +42x/84 +336/84= 84x
14x +7x +12x +420 +42x +336= 84x
14x +7x +12x +42x -84x= -420 -336
-9x = -756
x = -756/-9 =  84  years old.

 

 
 
Then we have that Diophantus was born about 200 and died about 284.

Thanks  Lidia for your effort!

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Lesson 7: Linear and Quadratic Equations. Maths Are Magic (V)

March 5, 2012

How old are you? 

If you ask this to your great-aunt it would be indiscreet. So we are going to do it in other way …Play the following game

Tell her:Think of a number between 1 and 9.
Multiply it by 9.
Take away this result from 10 times your age and tell me the answer. 

Now you could guess your great-aunt’s age.

 

You will always get your great-aunt’s age. Is it Magic or Maths?

Why are we always right? Could you give a logic reason?

Freely translated from Matemágicas. Descartes Project
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Lesson 7: Who Was Diophantus? Diophantus’s Riddle (IV)

February 25, 2012

Diophantus of Alexandria

 

Born: about 200   Died: about 284

Diophantus is often known as the ‘father of algebra’,but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics. Nevertheless, his remarkable, collection of problems is a singular achievement that was not fully appreciated and further developed until much later.

 He is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers. The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only six of the original 13 books were thought to have survived and it was also thought that the others must have been lost quite soon after they were written. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions.

 Diophantus did not use sophisticated algebraic notation, he did introduce an algebraic symbolism that used an abbreviation for the unknown and for the powers of the unknown.

 However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived.

You can read a good biography on

http://www.gap-system.org/~history/Biographies/Diophantus.html,

where I found this information. There are many biographies of famous mathematicians.

 Diophantus Riddle

 Much of our knowledge of the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. One of the problems (sometimes called his epitaph) states:

‘Here lies Diophantus,’ the wonder behold.

Through art algebraic, the stone tells how old:

‘God gave him his boyhood one-sixth of his life,

One twelfth more as youth while whiskers grew rife;

And then yet one-seventh ere marriage begun;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his father’s life chill fate took him.

After consoling his fate by the science of numbers for four years, he ended his life.’

 

This puzzle implies that Diophantus lived …..Could  you tell it to us?

You will publish a post with your solution on this blog and there is an interesting reward for you!

 However, the accuracy of the information cannot be independently confirmed.