Guitar Mathematics

Passy Playing iAxe Guitar
Image Copyright 2012 by Passy’s World of Mathematics

Here at Passy’s World we love playing guitar.

What is really cool is all of the mathematics involved with this amazing instrument.

In this lesson we look at the mathematics associated with the guitar in rock music.

Let’s start with the frets on the guitar neck. We use frets to play specific musical notes.

 
 

Pythagoras and Guitar Fret Spacings

Pythagoras was a Greek Mathematician who is famous for his mathematical analysis of the lengths of the sides of right angled triangles.

Pythagoras Triangles Diagram
Image Source: http://www.mathsisfun.com

However, Pythagoras worked on a lot of other mathematical ideas, including working out how long guitar strings need to be to create certain notes.

He found that mathematically, the note’s pitch is inversely proportional to the length of the string.

For example if we halve the length of the string, we create the exact same note, but one Octave higher.

This happens on a guitar when we play a note at the 12th fret. Positioning our finger at the 12th fret position, makes the string exactly half as long as its full length with no finger on any frets.

The Inverse Proportion means that if we play 1/2 of the string, we get 2 times the frequency of vibration of the string. This means the musical note gets “twice as big”, making it become the same note, but one octave higher.

Frequency (or how high the note pitch is) increases directly as the length of the string is decreased. This is the fundamental mathematics of all stringed instruments which Pythagoras figured out.

This is shown in the following diagram, along with other key string lengths that are created using the frets on a guitar.

Pythagoras Guitar
Image Copyright 2012 by Passy’s World of Mathematics

(Click the above picture to enlarge to full screen).

As can be seen in this diagram, Pythagoras worked out in ancient times that using frets to change the string length into certain fractions of the full length, resulted in certain pleasant sounding musical notes.

These Pythagoras fraction values are still used today when making guitars.

 

There is a great detailed article about Pythagoras and fret spacing on the Noyce Guitar page at the following link:

http://www.noyceguitars.com/Technotes/Articles/T4.html

Ian Noyce is a brilliant guitar maker, (called a “luthier”), right here in Melbourne Australia.

One day I would love to own a Noyce Bass Guitar, as a friend of mine has one, and it feels and sounds absolutely brilliant.

Check out the Noyce Guitar page at this web address:

http://www.noyceguitars.com/index.html

 
 

Guitar Notes Mathematics

Here is how the Pythagorean fret positions translate into actual musical notes on the guitar.

Guitar Notes on Fretboard
Image Source: http://www.guitar-chord.org

There sure are a lot of notes to memorise!

Thankfully there is a mathematical pattern that we can use to find notes on Guitar Strings.

First we need to assign numerical values to all of the notes in one octave of the Chromatic Scale as follows:

Assigning Numbers to Notes of Scale
Image Source: http://www.guitarnoise.com

In this mapping the natural notes are A-B-C-D-E-F-G = (0-2-3-5-7-8-10).

If we need to “sharpen” a note, we add 1, and if we need to “flatten” a note, we subtract 1.

The open strings on a guitar have the following numbers: E-A-D-G-B-E = 7-0-5-10-2-7.

To determine the note for any Fret we play on, we add the fret number and the string value to obtain a Total.

We then look this Total up in our Table (that we need to memorize), to work out the note.

For example, on the bottom string E (Note value = 7) then if we are at Fret 3, we do 7 + 3 = 10.

The number 10 in our table is the note “G”.

However, if we do the 7th fret of the bottom E string we get 7 + 7 = 14 which is past the end of our look up table.

Whenever we get a Total Answer greater than 11, we keep subtracting 12 from our Total Answer, until we get a number value between 0 and 11 that we can look up.

For our 7th fret E string example we do 14 – 12 = 2 which is the note “B”.

Working the opposite way, eg. working out the fret number for a note we want to play is a little bit trickier, but can be done.

To find out how, read the excellent article by Bruce Cyburt at the link below:

http://www.guitarnoise.com/lesson/from-math-to-music/

 
 

Mathematical Formulas for Frets

There are two Mathematical Formulas for obtaining the correct Pythagoras Fractions for Fret Spacings on the Guitar.

The first rule is commonly called the “Rule of 18”, but the actual value used is 17.817 .

This rule tells us how far the frets need to be apart from each other.

Using this rule we do the string length (from the zero fret “nut” at the top of the guitar to the bridge at the bottom of the guitar) divided by the value of 17.817 and this tells us how far the first fret is from the top “nut” zero fret of the guitar.

Trying this out on one of my own guitars produced 651mm / 17.817 = 36.6mm to the first fret.

To get the position of the second fret we now do (string length – distance to first fret) divided by the value of 17.817 .

This answer is then the distance from the first fret to the second fret.

We keep dividing up the string like this by 17.817 to get all of the fret positions relative to each other.

 

There is also a mathematical formula which uses powers of 2 in the denominator of the right hand side fraction.

This formula tells us how far each fret needs to be from the bridge at the bottom of the guitar.

This formula works as follows:

dist(x) = s / (2 ^ (x/12) )

where

x = number of the fret being evaluated

dist(x) = distance from bridge at the bottom end of the guitar to fret x on the neck

s = resting string length (distance from the zero fret “nut” at the top of the guitar to the bridge at the bottom of the guitar)

12 = the number of frets per octave that occurs on every Guitar.

This produces the positions for the standard “equal-tempered scale”.

The nut-to-fret positions and fret-to-fret positions are then produced by simple subtractions.

For example, on one of my Guitars the String Resting Length from Bridge to Nut is 651 mm.

If we put in x = 1 for the first fret, and do the calculation, then the answer is 614 mm.

Measuring with a tape measure on my guitar, this is correct, and the distance from the Bridge to the first fret is in fact 614 mm.

Subtracting 651 – 614 gives the distance from the top zero fret “nut” to the first fret to be 37mm.

Again measuring this on my actual guitar showed this 37mm answer to be correct.

Information Source: http://www.musemath.com/miscellany/ruleOf18.html

 
 

Vibration of Guitar Strings

The frequency (or size) of a vibrating guitar string depends on the length of the string being plucked, which we create by placing our fingers at various fret positions on the guitar.

These fret positions were originally devised by the ancient Greek Mathematician “Pythagoras”.

As we move further up the fretboard on a given string, we create shorter string length, and therefore higher pitched notes.

Each note on the guitar has a specific frequency of vibration, as shown in the following animated diagram for an E” note on the guitar:

Guitar String Vibration Animated GIF
Image Source: http://www.upscale.utoronto.ca

When we bend a string on the guitar, we can make it shift up a note, because we increase the value of the frequency by stretching the string. This is the same thing that happens when we tighten the machine heads to tune a given string.

 

As well as the main vibration of the string, we also get additional “overtone” sound wave vibrations.

For an “A” note 440 Hz fundamental vibration, the overtones occur at 2 × 440 = 880 Hz, 3 × 440 = 1320 Hz, 4 × 440 = 1760 Hz, 5 × 440 = 2200 Hz, 6 × 440 Hz = 2640 Hz, etc.

Overtone Vibrations
Image Source: http://www.upscale.utoronto.ca

The presence of overtones on any note played on an instrument gives the instrument its “timbre” or distinctive sound.

For example an A note played on a guitar sounds different to an A note played on a violin, due to the differences in overtone instensities.

This mathematics of what sound waves are produced, and in what intensities, makes a Violin sound like a Violin, and a Guitar sound like a Guitar.

For more detailed information on the mathematics of string vibrations, check out the following excellent web page created by David M. Harrison, Department of Physics, University of Toronto in Canada.

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Vibrations/Vibrations.html

(There is where we obtained the information for this part of the lesson).

 
 

Fibonacci Sequence in Music

Passy Plays Guitar
Image Copyright 2012 by Passy’s World of Mathematics

There is a set of numbers called the “Fibonacci Sequence” which are the values of 1,2,3,5,8 etc; and these values from the most important notes in the musical scale.

A scale is comprised of 8 notes, of which the 5th and 3rd notes create the basic foundation of all chords.

We also have harmonising intervals of 3rds and 5ths.

To produce a “third” we play the first note and a note 2 tones higher than this note.

To play an Octave we play the first note, and then a another note 8 tones higher.

To play a “Power Chord” for rock songs, we play the first note of a chord, and then another note 5 tones higher.

Note in all of the above, (which is the basis of rock music), all of the notes and intervals only use the Fibonacci Numbers 1, 2, 3, 5, and 8.

For more details about the Fibonacci Sequence in Music, click the following link to our separate detailed lesson about this.

http://passyworldofmathematics.com/fibonacci-sequence-in-music/

 
 

Octaves – Working in Eights

An Octave is basically the same note played 8 tones higher or lower from where you are currently on the fretboard. The number eight is a Fibonacci Number.

When two notes that are 8 notes apart are played at the same time, it can make a smooth Jazzy sound, or with overdrive, a really cool heavy metal sound.

If you are playing Octaves on Guitar using a pick, then you have to use some of your fingers to mute the strings in between. Depending on the shape you are doing, there might be one or two strings that require muting.

Here are the Octave forms that can be made on the guitar fret board:

One string in between (use a finger to mute this inbetween string, or if playing notes separately, skip this string) :

Octave Frets Diagram 1
Image Source: http://www.passyworld.com

Two strings in between Octave forms:

Octave Frets Diagram 2
Image Source: http://www.passyworld.com

Here is a video which shows how to use Octaves in Heavy Metal music.

 
 

Power Chords – Working in Fifths

Another Fibonacci Number, (apart from the number 8 used for Octaves), is the number 5.

The number 5 is also very important for playing rock guitar.

“Power Chords” are basically 2 or 3 notes that have been taken from a barre chord, and are played as a group on electric guitar, with distortion or overdrive liberally applied.

They are the basis of many rock songs, and musically they are based on 5th intervals.

Eg. We play only the first and fifth note of a chord, where 1 and 5 are both Fibonacci Numbers.

Here is a great eight minute video from YouTube about the basics of Power Chords:

 
 

Triplet Notes – Working in Fractions

Lead Guitarist Soloing
Image Source: http://www.passyworld.com

A lot of lead guitar soloing involves playing groups of three notes called “Triplets”.

These notes are played to the fraction values of the sound “Trip-o-Let”, and played fast enough to equal the equivalent time of either a quarter note or a sixtennth note.

Since “Trip-o-Let” has eight letters, the shortest note in the group of three syllables is the middle note which is 1/8 th of the Triplet.

The longest note in the triplet is the first note which is 4/8 or 1/2 the length of the triplet.

The final note, or the “Let”, is 3/8.

Knowing these fractions is important if synthesizing music using a Sequencer or Computerised Recorder, as we need to know how to “Quantise” the notes to make them the length of a real triplet.

Triplets are also used on Hi-Hats and other drums when programming drum machines, and so knowing their math is important for drummers as well as guitarists.

 

For example if we want to produce the sound of a medium speed triplet, then our whole Triplet needs to equal a quarter note.

This means that our “Trip-o-Let” needs to broken up into fractions of 1/4.

The exact time fractions they need to be are as follows:

Trip = 4/8 of 1/4 = 1/2 x 1/4 = 1/8 th.

O = 1/8 of 1/4 = 1/8 x 1/4 = 1/32 nd

Let = 3/8 of 1/4 = 3/8 x 1/4 = 3/32 nd.

 

To create Triplets on the Guitar, we have to use special playing techniques called “Hammer-Ons” and “Pull-Offs”.

Here is a great video demonstrating the use of Triplets in Guitar soloing.

 
 

Mathematics of Songs – Working in Fours

Most Rock songs are played in 4/4 time, where there are four beats to very bar of music.

We then have 4 bar, 8 bar, and 16 bar sections in songs, as well as the classic “12 Bar Blues”.

As can be seen the whole song is based around multiples of four, and this is the heart of rock music, as groups of four are very pleasing to the human ear and brain.

The use of groups of four is essential when using Computer Music Composition Programs such as Apple’s Garage Band, or Magix Music Maker.

Magix Music Maker
Image Source: Screen Capture of Magix Music Maker 15

 
 

Mathematical Patterns in Songs

Passy Singing with Band
Image Copyright 2012 by Passy’s World of Mathematics

There are a few very common patterns which are used to construct most of the songs that are played on Guitar.

Vocalists need to know these patterns, and be able to count in twos, fours, and eights to know when they need to start singing at the beginning of a song, or after a solo has completed.

Guitarists also need to know the patterns so that they can change from Verse Chords to Chorus Chords, especially after a solo finishes.

Song structures are very mathematical and involve repeating Verses and Choruses one after each other.

One of the most commonly used formulas is the “Verse-Chorus-Bridge” song.

The overall song structure is as follows:

“Verse/Chorus/Verse/Chorus/Bridge/Chorus”

The Bridge is a different musical part to the Verse or Chorus, and can be either a musical solo or some continued vocals.

 

Another popular song formula is “Verse Chorus” which looks like this:

“Verse/Chorus/Verse/Chorus/Solo/Chorus”

In this song, the Bridge is replaced with a Musical Solo that is played over the top of one or two Verses of music with no vocals.

 

Less common formulas for songs are ones consisting of just Verses, and no Chorus; as well as the “Verse/Verse/Bridge/Verse” structure.

 
 

Jared Leto’s Pythagoras Guitar

McSwain Pythagoras Guitar
Image Source http://www.mcswainguitars.com

Jared Leto is an Actor and also lead singer of the band “30 Seconds to Mars”. One of his guitars is a white guitar called “Pythagoras”.

This guitar was custom made for him by McSwain Guitars who make brilliant customised guitars for a range of famous guitar players such as Slash from Guns and Roses.

Check out McSwain’s awesome Guitars at the following link:

http://www.mcswainguitars.com

One of Jarred’s band’s best songs is “Closer to the Edge” as featured in the following clip:

 
 

Building a “Pythagoras” Guitar Video

There is a lot of mathematics involved with building a guitar. The components need to be precisely measured and fitted to create a good “action” distance of the strings from the fretboard, as well as setting pickup distances, and getting the electronics connected correctly.

The following video is by a guy, who with his father, built a very nice copy of Jared Leto’s “Pythagoras” Guitar.

For those with a technical bent, the specifications for this guitar are as follows:

Seymour Duncan Invader Pickups
Sperzel Locking Tuners
Alder Explorer Neck Through Body
Gracies White Guitar Paint
500K pots

 

Another excellent video about the making of guitars is this one about the Australian Company: Maton Guitars.

It goes into the Mathematics of the Guitar, and features Ian Noyce talking about Fret Spacings.

 
 

Related Items

Fibonacci Sequence in Music
Playing Hammer On Guitar Notes
Playing Pull Off Guitar Notes
Playing Lead Guitar
How to Play Hendrix Hey Joe
How Guitar Pedals Work
Jazzy Guitar Octaves
Guitar Arpeggios
Playing Guitar Power Chords
Magix Music Maker
Adjusting Electric Guitars
Computer Enhanced Music
V-Guitar Free Online Lessons

 

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Angles and Parallel Lines

Getty Images Crooked Highway
Image Source: Licence pending from Getty Images

The “Crooked Mile” on the Pillani Highway in Maui Hawaii looks very different to a normal highway !

Volcanic activity and continual movement of the island makes it impossible to build a straight road that will last very long.

Highways are usually made much easier to drive on, by making both sides fo the road follow the exact same straight line direction. We say that the edges of the road are “parallel” to each other.

Highways also need to have the guard rails that run parallel to the surface level of the road, and all line markings also need to be done in parallel.

Parallel Lines in Road Marking
Image Source: http://constructionfield.net

Parallel Lines are of critical importance when marking out roads, pedestrian crossings, car parks, and airport runways.

Parallel Lines in Construction
Image Source: Home Designer Software

Parallel Lines are also vital on basketball, tennis, volleyball, netball, badminton, and squash courts, as well as on atheletics tracks.

Parallel Line Basketball Markings
Image Source: Home Designer Software

Parallel Lines are of critical importance in Landscape Design, Timber Deck Work, and Brick work. If all edges are not exactly parallel, then the construction job lacks quality.

Parallel Lines in Garden Design
Image Source: http://www.felmiatika.com

Train and Tram Tracks need to have rails which run perfectly parallel to each other.

Yellow Tram in Melbourne
Image Source: http://1.bp.blogspot.com

Electric Power Lines need to run in parallel so that their cables cannot touch each other and short circuit the power grid.

Multi-storey floors and rows of windows in high rise Buildings need to run in parallel.

Sets of Pipes and Cabling in buildings, ships, cars, and aeroplanes are also ran in parallel.

Strings and Frets on guitars and other musical instruments need to run exactly in parallel.

Passy Playing iAxe Guitar
Image Source: Copyright 2010 Passy’s World of ICT

 

In this lesson we look at the angle properties associated with parallel lines.

 
 

Definition of Parallel Lines

Parallel Lines are two or more lines that are always the same distance apart.

We place arrows on the lines to indicate that they are going in the same direction.

Parallel Lines Angles One
Image Copyright 2012 by Passy’s World

Often with Parallel components, there is also a linear item joining them, which is not at 90 degrees.

This occurs for example in car suspension, where the springs and shock absorbers connect at an angle to their parallel joining components.

Parallel Suspension transversal
Image Source: http://accurate-alignment.com

In Mathematics we draw this diagram as follows, which results in a set of eight angles.

Parallel Lines Angles Two
Image Copyright 2012 by Passy’s World

For the set of eight angles there are four common pairings which we use in mathematical Geometry.

These four pairs of Angles are known as:

Vertical “X” Angles

Alternate “Z” Angles

Corresponding “F” Angles

Co-Interior “C” Angles

In the sections which follow, we examine each of these four types of Parallel Lines Angles.

 
 

Vertical Angles

These are the pairs of angles which can be found in an “X” shape arrangement in any pair of Parallel Lines that are connected by a Transversal.

These angles are always equal in size to each other.

In mathematics we say they are “Congruent” Angles, because they have exactly the same size and shape.

Parallel Lines Vertical Angles
Image Copyright 2012 by Passy’s World

 
 

Alternate Angles

These are the pairs of angles which can be found in a “Z” shape arrangement in any pair of Parallel Lines that are connected by a Transversal.

Parallel Lines Alternate Angles
Image Copyright 2012 by Passy’s World

The “Z” shape can also be back to front. Either way around, “Z” type angles are always equal to each other in size.

In mathematics we say they are “Congruent” Angles, because they have exactly the same size and shape.

Parallel Lines Alternate Angles Two
Image Copyright 2012 by Passy’s World

 
 

Corresponding Angles

These are the pairs of angles which can be found in an “F” shape arrangement in any pair of Parallel Lines that are connected by a Transversal.

These angles are always equal in size to each other.

In mathematics we say they are “Congruent” Angles, because they have exactly the same size and shape.

Examples of these are shown in the following diagram.

Parallel Lines Corresponding Angles
Image Copyright 2012 by Passy’s World

 
 

Co-Interior Angles

Unlike Vertical, Alternate, and Corresponding Angles which are equal to each other; Co-Interior Angles are never equal to each other.

Co-Interior Angles exist in a “C” shape and do NOT equal equal each other.

However they always ADD up to equal 180 degrees. Because the sum to be 180, they are “Supplementary Angles”.

Parallel Lines Co-Interior Angles
Image Copyright 2012 by Passy’s World

 
 

Parallel Lines Videos

The following videos explain the proerties of the following angle types, as well as giving example questions and their solutions.

Vertical “X” Angles, Alternate “Z” Angles, Corresponding “F” Angles and Co-Interior “C” Angles

 

Vertical Angles Video

 
 

Alternate Angles Video

 
 

Corresponding Angles Videos

 
 

 
 

Co-Interior Angles Video

 
 

Parallel Lines Summary Video

The following video contains a complete overview of both Parallel and Perpendicular Lines

 
 

Examples of Parallel Lines Angles

We can use the Angle Properties of Parallel Lines to solve geometry questions as shown in the following examples.

The steps are basically the same for each question.

– Look carefully at the given angle, and one of the unknown variable angles, and see if they form one of the common patterns such as X-Shape, Z-Shape, F-Shape, and C-Shape.

– Mark the shape onto the parallel lines diagram.

– Use the properties to decide if the unknown angle is equal to the given angle, ( or if “C-Shape” is equal to 180 – the given angle ).

Remember that “C-Shape” angles are the annoying exception where the angles are Supplementary rather than Equal.

Parallel Lines Angles Examples One
Image Copyright 2012 by Passy’s World

 

Shown below are the solutions to Example 1.

 

Parallel Lines Angles Example One Answers
Image Copyright 2012 by Passy’s World

 

In the next set of examples, we have some Parallel Lines within shapes, and some have more than one relationship to deal with when solving the question.

Parallel Lines Angles Examples Two
Image Copyright 2012 by Passy’s World

 

Shown below are the solutions to Example 2.

 

Parallel Lines Angles Example Two Solutions
Image Copyright 2012 by Passy’s World

 
 

Parallel Lines Online Activity

Parallel Lines Online Activity

Watch the animated introduction and then do the online activity by either clicking on the picture or the following link.

http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/parallellinesact.shtml

 
 

Parallel Lines Online Quizes

Do the following quick online quiz that has three parallel lines questions.

http://www.bbc.co.uk/apps/ifl/schools/gcsebitesize/maths/quizengine?quiz=parallellines&templateStyle=maths

 

The following online quiz from Kahn Academy includes fully worked solutions for each question, and includes Algebra Angle questions.

Click the following link to do this Quiz.

http://www.khanacademy.org/math/geometry/parallel-and-perpendicular-lines/e

 
 

Related Items

Classifying Triangles
Angle Sum in a Triangle
Exterior Angle of a Triangle
GeoGebra
Interactives at Mathwarehouse
Jobs that use Geometry

 

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Exterior Angle of a Triangle

3D Triangle Drawing by Justin Windle
Image Source: http://www.soulwire.co.uk

Triangles can even be used in Art, as shown above in the great piece of artwork by Justin Windle from Soulwire.

What we really like about this picture is the clever shading of the triangles to create various 3D effects throughout the drawing.

 

Triangles are also used in GPS Location and Navigation.

Any position on Earth that is beamed to three or more GPS Satellites creates unique distances which form triangles of known sizes.

GPS Satellite Triangles
Image Source: http://www.cdxetextbook.com

The mathematical process to work out the actual GPS location on earth uses “Triangulation” or “Trilateration”.

Basically there are 24 GPS satellites orbiting the earth in fixed positions, so that any GPS unit will always be able to connect to at least three of the satellites at any time of the day or night.

From these radio frequency connections, many triangles are formed, calculations performed, and the latitude, longtitude, and elevation, of the position on earth is determined.

The actual mathematics is quite complicated, and uses 3D triangles, which form cone like Pyramids, and this is called “Trilateration”.

3D Diagram of GPS Satellites
Image Source: http://www.technogps.com

For an excellent overview of exactly how GPS works, check out the following web page:

http://www.cdxetextbook.com/auxil/periph/satellite/triangulation.html

In this lesson, we are not looking at the details of GPS Triangles, but rather the Exterior Angles which are formed when we extend the sides of any Triangle.

Exterior Angles can be used as part of GPS Calculations, but this mathematics is quite a bit more advanced than the basics we need to cover here.

 
 

Exterior Angles of a Triangle

The Exterior Angles of a Triangle are formed by extending the sides of the Triangle, so that a 180 degree Supplementary Angle straight line is formed.

This is shown in the following diagram.

Triangle Exterior Angles Supplementary Angles
Image Copyright 2012 by Passy’s World

Every Triangle has three Exterior Angles that match up with each of the three Interior Angles.

Each Exterior Angle sums with its adjacent Interior Angle to form a 180 degree straight line.

The following example shows how we extend a typical triangle’s sides to create its three Exterior Angles.

Triangle Three Exterior Angles
Image Copyright 2012 by Passy’s World

 
 

Exterior Angle Theorem

Based on the fact that the Interior Angles of all triangles add up to 180 degrees, and that the Exterior Angle and its partner angle also always add to 180 degrees, Mathematicians have been able to develop the rule shown in the diagram below.

They call this rule a “Theorem”, which is just a fancy name for any shortcut rule we can use in Maths.

Triangle Exterior Angles Theorem
Image Copyright 2012 by Passy’s World

 
 

Exterior Angle of Triangle Examples

In this first example, we use the Exterior Angle Theorem to add together two remote interior angles and thereby find the unknown Exterior Angle.

This is the simplest type of Exterior Angles maths question.

Triangle Exterior Angles Example 1
Image Copyright 2012 by Passy’s World

 

In this next example we find the missing Interior Angle by using the Exterior Angle Theorem.

Triangle Exterior Angles Example 2
Image Copyright 2012 by Passy’s World

 

This final example shows an Exterior Angle question that requires the setting up and solving of Algebra Equations.

Triangle Exterior Angles Example 3
Image Copyright 2012 by Passy’s World

 
 

Videos About Exterior Angles

The following video from YouTube shows how we use the Exterior Angle Theorem to find unknown angles.

 
 

Here is another video which shows how to do typical Exterior Angle questions for triangles.

 
 

This video shows some examples that require algebra equations to solve for missing angle values.

 

 

Exterior Angle of a Triangle Worksheets

The links below are to web pages which have a range of questions involving exterior angles.

http://www.mathwarehouse.com/classroom/worksheets/triangles/remote-exterior-interior-angles.pdf

http://classroom.westsidehsfaculty.org/webs/jschroe1/upload/worksheet_triangle_sum_and_exterior_angle.pdf

 
 

Related Items

Classifying Triangles
Angle Sum in a Triangle
GeoGebra
Interactives at Mathwarehouse
Jobs that use Geometry

 

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Angle Sum in a Triangle

House Frame with Triangles
Image Source: http://www.greenpest.com.au

Building Frames are “triangulated” to give them strength. Without a triangle structure the wood frame will not be rigid or strong.

In Australia, the Sydney Harbor Bridge is made of an interwoven triangle structure.

Sydney Harbour Bridge Triangles
Image Source: http://www.orangehelicopters.com.au

Every part of the bridge is extensively triangulated to create rigidity and strength.

Sydney Harbor Bridge triangles Detail
Image Source: http://www.dinkumaussies.com

Builders of Structural Frames and Steel Bridges need to have an intimate knowledge of angles and triangles.

In this lesson we look at how the three angles in any triangle always create a total of 180 degrees.

 
 

Sum of the Angles in a Triangle

The three angles in any triangle always create a total of 180 degrees.

Two mathematical proofs for angles adding up to 180 degrees in any triangle are shown in the following video.

The first proof involves hands on cutting up a paper triangle, and the second one is a far more technical geometric proof.

 

The following shows the 180 degrees total rule for several different triangles.

Angle Sum of Triangle
Image Copyright 2012 by Passy’s World

 
 

Videos About Triangle Angles

This first video shows several different examples of how to find missing Angles in Triangles.

 
 

This next video shows a more complex problem involving Ratios, and how to use an Algebra Equation to solve for the unknown angles.

 
 

Here is another video that involves using Algebra to find unknown angles.

 
 

Finding an Unknown Angle in a Triangle

Because we know that the three angles add up to 180 degrees, we can easily work out the value for a missing angle.

When a question gives us the values for two interior angles of a triangle, the third missing angle is always 180 minus the two angles that we know.

Missing Angle = 180 minus the other two given angles

The following examples show how to calculate the missing angle for several different triangles, using Algebara equations.

Triangle Sum Examples
Image Copyright 2012 by Passy’s World

 
 

Math Warehouse Triangle Interactive

Math Warehouse Triangle Interactive

The middle of the following web page has a drag around interactive which shows the angles always adding to 180, no matter waht shape we make the triangle.

(Much quicker than cutting out lots of random triangles from paper and measuring them).

Note that it is best to set the units button to “Integer” whole numbers when using this demonstration tool.

Click the following link to use this interactive

http://www.mathwarehouse.com/geometry/triangles/

 
 

Finding Angles Worksheet

Click the following link to get a PDF worksheet on finding Triangle Angles.

http://www.mathworksheets4kids.com/triangles/missing-angles.pdf

 
 

Finding Angles Game

Tank Attack Angles in Triangles Game

“Tank Attack” has some quite challenging questions on Supplementary Angles, as well as Triangles.

Click the following link to play this game.

http://www.what2learn.com/home/examgames/maths/angles1/

 
 

Solving Angles Game

Itzi Spider Angles Game

This game about “Itzi” spider involves solving several levels of Angles questions, where later levels include triangles.

Click the following link to play this challenging game.

http://www.mangahigh.com/en_au/games/atangledweb?localeset=en_au

 
 

Related Items

Classifying Triangles
GeoGebra
Interactives at Mathwarehouse
Jobs that use Geometry

 

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Classifying Triangles

Triangular Structure of Beryl Gem
Image Source: http://www.dalamatiacity.com

The above diagram shows the atomic structure inside the crystal gemstone “Beryl”, which is composed of Silicon, Aluminium, Beryllium, and Oxygen ions.

Topaz, Olivine, Amethyst, and several other crystals found in nature also have triangular atomic structures.

The triangle is a natural shape which gives great strength to any structure.

Nature uses Triangles to make strong solid materials, which due to their internal geometry can be cut into beautiful gem stones.

Humans use Triangles to create strength in structures such as High Voltage Electricity Towers, structural frames for building and bridges, and even on farm gates.

worker and high voltage tower
Image Source: http://www.nationalgrid.com

 
 

Types of Triangles

A triangle is a geometrical shape made up of 3 points called “Vertices” which are connected together by lines called “Segments”.

Eg. A triangle has three sides and is made of straight lines.

A triangle may be classified by how many of its sides are of equal length.

It can also be classified by what types of angles it has.

The vertices are labelled using upper case capital letters like A B and C, or D, E, and F, or P, Q, and R and so on.

The Angles inside the triangle which correspond to these Vertex points are labelled using lower case letters such as a,b,c or d,e,f and so on.

The line segments are labelled according to the vertex points that they connect to, such as AB, BC, or AC. Note that we keep the letters in alphabetical order, and also write lines above them to indicate that they refer to the linear sides of the triangle.

This is summarised in the following diagram.

Triangle Types - Classifying Triangles 1
Image Copyright 2012 by Passy’s World

 

There are four main types of Triangles that we use in the mathematics of geometry.

These are called “Equilateral”, “Isosceles”, “Scalene”, and “Right Angled” triangles.

The properties of these four types of triangles are shown in the following diagrams.

Triangle Types - Classifying Triangles
Image Copyright 2012 by Passy’s World

Triangles are also referred to as being either “Acute”, “Obtuse”, or “Right” Triangles.

“Acute” triangles have all three of their angles less than 90 degrees in size, such as the Equilateral and Isosceles triangles shown ablove.

“Obtuse” triangles contain one angle which is larger than 90 degrees, like the Scalene triangle shown above.

A “Right” Triangle that contains one L-shaped 90 degree angle is neither Acute or Obtuse. It is exactly inbetween these two, and is given the special name of “Right Triangle”.

This is summarised in the following diagram.

Classification of Triangles by Sides and Angles
Image Source: http://mrferrell.pbworks.com

 
 

Videos About Classifying Triangles

The following video explains how to classify triangles as Acute, Obtuse, or Right.

 
 

This next video is about Equilateral, Isosceles, and Scalene triangles.

 
 

Here is a great rocking music clip all about triangles.

Note that the word “Congruent” means “Identical” or “Equal”.

 
 

Classifying Triangles Worksheets

There are a number of different PDF worksheets, with answers available for free at the following links:

http://www.mathworksheets4kids.com/triangles.html

http://www.k6-geometric-shapes.com/triangle-worksheet.html

http://www.helpingwithmath.com/printables/worksheets/geo0701triangles01.htm

 
 

Classifying Triangles Games

Classifying Triangles Game 1

This first game involves getting Baseball hits everytime you classify a given triangle correctly.

Make sure you click on the right hand corner question mark to hear the instructions about how to swing at the ball.

Click the following link to play this fun game.

http://www.factmonster.com/math/knowledgebox/player.html?movie=sfw41507

 
 

Classifying Triangles Game 2

In the above game we use the mouse to grab the triangles as they pass by, and place them into the correct basket.

Click the following link to play this game.

http://www.math-play.com/classifying-triangles/classifying-triangles.html

 
 

Classifying Triangles Game 3

This next game from BBC Mathematics is similar to the previous game, except that it focusses on Equilateral, Isosceles, and Scalene triangles, as well as some common four sided figures called “Quadrilaterals”.

Click the following link to play this game.

http://www.bbc.co.uk/bitesize/ks2/maths/shape_space/shapes/play/

 
 

Related Items

Geometry Interactives at Geogebra
Interactives at Math Warehouse
Jobs that use Geometry

 

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Passy

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Posted in Geometry, Triangles | Tagged , , , , , , , , , , , , , , | 16 Comments