Polyhedron / Poliedros

A polyhedron is a solid with flat faces
(from Greek poly- meaning "many" and -edron meaning "face").
Each face is a polygon (a flat shape with straight sides).

Cuando hablamos de figura geométrica, lo hacemos en dos dimensiones (2D): largo y ancho, pero cuando hablamos de un cuerpo geométrico lo hacemos en tres dimensiones (3D) largo, ancho y alto. Es así como dejamos de hablar de una geometría plana, para introducirnos en el estudio de una geometría espacial. En la figura podemos observar, gráficamente, la diferencia entre ambos objetos matemáticos.

 Los poliedros se relacionan estrechamente con las figuras planas, ya que un cuerpo geométrico está compuesto por caras que tienen forma de figuras planas o polígonos, como normalmente las conocemos. Es así entonces que definimos a los poliedros de la siguiente manera: Un poliedro es la región del espacio delimitada por polígonos. Lo poliedros o sólidos geométricos son figuras en tres dimensiones que poseen dos propiedades: el área de la superficie y el volumen.  Los cuerpos geométricos se dividen en cuerpos redondos y poliedros.

Podemos reconocer una anatomía de los poliedros así:

Tomado de: http://cpdiacfisicoquimico.blogspot.com

• Cara: cada uno de los polígonos que forman  el poliedro
• Aristas: Línea que se forma donde se unen dos caras
• Vértices: Punto donde se cortan tres aristas.
• Ángulos diedros: Ángulo formado por dos caras que tienen una arista en común.
• Ángulo poliedro: Ángulo formado por tres o más caras que tienen un vértice en común.
• Diagonal: Recta que une dos vértices  de caras diferentes
•  

Three Dimensions
It is called three-dimensional, or 3D because there are three dimensions: width, depth and height

Quedan por nombrar otros elementos que se irán estudiando con el paso del tiempo y que tienen que ver con algunas condiciones particulares de cada grupo de cuerpos geométricos.

Miremos ahora una breve clasificación de poliedros. Empecemos por los más estudiados que son los poliedros regulares o sólidos platónicos, recordemos que se llaman así porque todas sus caras son polígonos regulares.

 Los primas y pirámides reciben sus nombres según la forma poligonal de sus bases, miremos algunas de ellas

Luego de mirar los anteriores cuerpo geométricos observemos un poco sobre qué es o qué son los cuerpos redondos. Estos son sólidos limitados por regiones curvas o planas y curvas, es decir, son aquellos que tienen al menos una de sus caras de forma curva; también se llaman cuerpos de revolución porque pueden obtener a partir de una figura alrededor de un eje.

Counting Faces, Vertices and Edges

When we count the number of faces (the flat surfaces), vertices (corner points), and edges of a polyhedron we discover an interesting thing:

The number of faces plus the number of vertices
minus the number of edges equals 2

This can be written neatly as a little equation:

F + V − E = 2

It is known as Euler's Formula (or the "Polyhedral Formula") and is very useful to make sure we have counted correctly!

Let's try some examples:

This cube has: 6 Faces 8 Vertices (corner points) 12 Edges
F + V − E = 6 + 8 − 12 = 2

This prism has: 5 Faces 6 Vertices (corner points) 9 Edges
F + V − E = 5 + 6 − 9 = 2

But there are cases where it does not work! Read Euler's Formula for more



Otra clase de sólidos de importancia son los llamados Arquimedianos o sólidos truncados, estos cuerpos son el resultado de un truncamiento o corte transversal en los vértices de los sólidos platónicos, existen dos tipos de corte: a un tercio o a un medio. Miremos algunos de ellos.

Para terminar es recomendable que veamos el siguiente video que nos permitirá entender, un poco más a fondo, sobre qué son los poliedros y los cuerpos redondos.

 
Todos los videos, aplicaciones e imágenes pertenecen a sus autores, en este sitio web se usarán, respetando la autoría de ellos y con fines meramente educativos, no lucrativ

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Sexagesimal system

     Sexagesimal is a numeral system in which each unit is divided into 

60 units of lower order, that is to say, it is a base-60 number system

The sexagesimal system was used by the Sumerians and Babylonians.

 It is currently used to measure time and angles

1 h 60 min 60 s

1º 60' 60''

Converting Sexagesimal into Decimal Form

Convert 3 hours, 36 minutes, 42 seconds to seconds.

Converting Decimal into Sexagesimal Form

1To convert to major units, divide.

7,520''

2To convert to minor units, multiply.

For the measurement of time and angles, the following steps can be performed:

Addition

1. Place the hours under the hours (or the degrees under the degrees), the minutes under the minutes and the seconds under the seconds and add together.

2. If the seconds total more than 60, they are divided by 60, the remainder will remain in the seconds column and the quotient is added to the minutes column.

3. Repeat the same process for the minutes.

Subtraction

1. Place the hours under the hours (or the degrees under the degrees), the minutes under the minutes and seconds under seconds and subtract.

2. If it is not possible to subtract the seconds, convert a minute of the minuend into 60 seconds and add it to the minuend seconds. Then, the subtraction of the seconds will be possible.

3. Repeat the same process for the minutes.

Multiplication by a Number

1. Multiply the seconds, minutes and hours (or degrees) by number.

2. If the seconds exceed 60, divide that number by 60, the remainder will remain in the the seconds column and the quotient is added to the minutes column.

3. Repeat the same process for the minutes.

Division by a number 

1. Divide the hours (or degrees) by the number.

Divide 37º 48' 25'' by 5.

2. The quotient becomes the degrees and the remainder becomes the minutes when multiplied by 60.

3. Add these minutes to the minutes column and repeat the same process for the minutes.

4. Add these seconds to the seconds column and then divide the seconds by the number.

Worksheet

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Equations

What is an Equation

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equation says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

Parts of an Equation

So people can talk about equations, there are names for different parts (better than saying "that thingy there"!)

Here we have an equation that says 4x − 7 equals 5, and all its parts:

A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.

A number on its own is called a Constant.

A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)

Sometimes a letter stands in for the number:

Example: ax² + bx + c

• x is a variable
• a and b are coefficients
• c is a constant

An Operator is a symbol (such as +, ×, etc) that shows an operation (ie we want to do something with the values).

A Term is either a single number or a variable, or numbers and variables multiplied together.

An Expression is a group of terms (the terms are separated by + or − signs)

So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"

Writing Algebraic Equations

Problem:	Jeanne has $17 in her piggy bank. How much money does she need to buy a game that costs $68?
Solution:	Let x represent the amount of money Jeanne needs. Then the following equation can represent this problem:
	17 + x = 68
	We can subtract 17 from both sides of the equation to find the value of x.
	68 - 17 = x
Answer:	x = 51, so Jeanne needs $51 to buy the game.

In the problem above, x is a variable. The symbols 17 + x = 68 form an algebraic equation. Let's look at some examples of writing algebraic equations.

Example 1:	Write each sentence as an algebraic equation.
	Sentence Algebraic Equation A number increased by nine is fifteen. y + 9 = 15 Twice a number is eighteen. 2n = 18 Four less than a number is twenty. x - 4 = 20 A number divided by six is eight.

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Example 2:	Write each sentence as an algebraic equation.
	Sentence Algebraic Equation Twice a number, decreased by twenty-nine, is seven. 2t - 29 = 7 Thirty-two is twice a number increased by eight. 32 = 2a + 8 The quotient of fifty and five more than a number is ten. Twelve is sixteen less than four times a number. 12 = 4x - 16

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Example 3:	Write each sentence as an algebraic equation.
	Sentence Algebraic Equation Eleni is x years old. In thirteen years she will be twenty-four years old. x + 13 = 24 Each piece of candy costs 25 cents. The price of h pieces of candy is $2.00. 25h = 200 or .25h = 2.00 Suzanne made a withdrawal of d dollars from her savings account. Her old balance was $350, and her new balance is $280. 350 - d = 280 A large pizza pie with 15 slices is shared among p students so that each student's share is 3 slices.

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Summary:

An algebraic equation is an equation that includes one or more variables. In this lesson, we learned how to write a sentence as an algebraic equation. Resolviendo ecuaciones, para practica

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Proportionality

Proportions

To practice1

To practice2

Proportion says that two ratios (or fractions) are equal.

Example:

 

So 1-out-of-3 is equal to 2-out-of-6

The ratios are the same, so they are in proportion.

Example: Rope

A ropes length and weight are in proportion.

When 20m of rope weighs 1kg, then:

• 40m of that rope weighs 2kg
• 200m of that rope weighs 10kg
• etc.

So:

201 = 402

Sizes

When shapes are "in proportion" their relative sizes are the same.

Here we see that the ratios of head length to body length are the same in both drawings. So they are proportional. Making the head too long or short would look bad!

1.   What does it mean to say that two quantities are directly proportional (or simply, proportional)? By whatever ratio one quantity changes, the other changes in the same ratio.

For example, let us say that the distance you travel is proportional to the time.  This means that if you travel twice as long, you will go twice as far.  If you travel three times as long, you will go three times as far.  While if you travel half as long, you will go half as far. By whatever ratio the time changes, the distance will change proportionally, that is, in the same ratio.

2.   How do we solve problems of proportionality? Form ratios between the things of the same kind.

To begin with, we can only form ratios between things of the same kind:  length to length, time to time, dollars to dollars, and so on.  A ratio between things of different kinds ("This amount of money is half of this amount of time") makes no sense. When we relate things of different kinds, as in "dollars per (for each) hour," that is not called a ratio but arate. Example 1.   In 4 hours, you can travel 110 miles.  How far can you travel in 8 hours? Answer.  Since you travel twice as many hours -- 8 hours are twice as many as 4 hours -- then you will travel twice as many miles. 2 × 110 miles = 220 miles. Proportionally, upon forming ratios between things of the same kind: 4 hours are to 8 hours  as  110 miles are to 220 miles. Example 2.   Maria can earn $70 in 6 hours.  How much will she earn in 18 hours? How much in 3 hours? How much in 27 hours? Solution.  Earnings are directly proportional to time.  If Maria worksthree times as long -- 18 hours -- then she will earn three times as much.  She will earn 3 × $70 = $210. As for 3 hours, they are half of 6 hours: therefore she will earn halfof $70; she will earn $35. Finally, 27 hours.  Every 6 hours she earns $70.  How many 6 hours are there in 27? 27 hours = 24 hours + 3 hours. In 24 hours there are four 6's. She will earn 4 × $70 = $280. In 3 hours, we have seen that she will earn $35. Therefore, in 27 hours she will earn $280 + $35 = $315. Equivalently, 27 hours are four and a half times6 hours.  Therefore she will earn four and a half times $70. When two quantities are directly proportional, we say that one of them varies directly as the other.  In this example, wages vary directly as time. Example 3.   Which is a better value: 12 ounces for $5.00, or 48 ounces for $22.00? Answer.  48 ounces are four times 12 ounces.  Therefore we should expect the price to be four times more.  But four times $5 is $20 -- which is less than $22.  Therefore 12 ounces for $5 is a better value. Example 4.   Which is a better value: 15 ounces for $9.00, or 20 ounces for $11.00? Answer.  20 ounces are 5 ounces more than 15 ounces.  That is, 20 ounces are one third more.  Now, one third more than $9.00 is $9.00 + $3.00 = $12.00 Therefore, 20 ounces for $11.00 is a better value.  You save $1.00. Inverse proportionality

3.   What does it mean to say that two quantities are inversely proportional? By whatever ratio one quantity changes, the other changes in the inverse ratio.

This means that if one of the quantities doubles, then the other will become half as large.  For the inverse of the ratio 2 to 1 ("doubles") is the ratio 1 to 2 ("half").  The terms are exchanged. Example 5.   Let us suppose that the time it takes to do a job is inversely proportional to the number of workers.  The more workers, the shorter the time. Specifically:  If 6 workers can do a job in 4 days, then how long will it take 12 workers? Answer.  The number of workers has doubled , going from 6 to 12.  Therefore it will take only half as many days.  It will take only 2 days. Example 6.   The speed that a car can achieve in 10 seconds is inversely proportional to its weight.  (That is, the more the car weighs, the slowerit will be going.) After 10 seconds, a car that weighs 2400 pounds can achieve a speed of 44 miles per hour.  If the car weighed 1600 pounds, how fast would it be going? Answer.  What ratio has the new weight to the original weight -- 1600 pounds to 2400 pounds?  1600 is two thirds of 2400: 1600 is to 2400  as  16 is to 24  as  2 is to 3.

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