How to Find the Probability of an Event and Calculate Odds, Permutations and Combinations

Learn about probabilities, odds, permutations and combinations. blickpixel, public domain image via Pixabay

What Is Probability Theory?

Probability theory is an interesting area of statistics concerned with the odds or chances of an event happening in a trial, e.g., getting a six when a dice is thrown or drawing an ace of hearts from a pack of cards. To work out odds, we also need to have an understanding of permutations and combinations. The math isn't terribly complicated, so read on and you might be enlightened!

What's covered in this guide:

• Equations for working out permutations and combinations
• Expectation of an event
• Addition and multiplication laws of probability
• General binomial distribution
• Working out the probability of winning a lottery

Definitions

Before we get started let's review a few key terms.

• Probability is a measure of the likelihood of an event occurring.
• A trial is an experiment or test, e.g., throwing a dice or a coin.
• The outcome is the result of a trial, e.g., the number when a dice is thrown, or the card pulled from a shuffled pack.
• An event is an outcome of interest, e.g., getting a 6 in a dice throw or drawing an ace.
• Odds is the probability of an event occurring divided by the probability of it not occurring (e.g., 1 to 5 chance of a six in a dice throw).

What Is the Probability of an Event?

There are two types of probability, empirical and classical.

If A is the event of interest, then we can denote the probability of A occurring as P(A).

Empirical Probability

This is determined by carrying out a series of trials. So, for instance, a batch of products is tested and the number of faulty items is noted plus the number of acceptable items.

If there are n trials

and A is the event of interest

Then if event A occurs x times

P(A) = x / n

Empirical Probability Example

A sample of 200 products is tested and 4 faulty items are found. What is the probability of a product being faulty?

So x = 4 and n = 200

Therefore P(faulty item) = 4 / 200 = 0.02 or 2%
If we do the trial again with a different number of products, we can expect 2% of them to be faulty.

Classical Probability

This is a theoretical probability which can be worked out mathematically.

If A is the event, then

P(A) = Number of ways the event can occur / The total number of possible outcomes

Example 1

What are the chances of getting a 6 when a dice is thrown?

In this example, there is only 1 way a 6 can occur and there are 6 possible outcomes, i.e. 1, 2, 3, 4, 5 or 6.

Scroll to Continue

So P(6) = 1/6

Example 2

What is the probability of drawing a 4 from a pack of cards in one trial?

There are 4 ways a 4 can occur, i.e., 4 of hearts, 4 of spades, 4 of diamonds or 4 of clubs.

Since there are 52 cards, there are 52 possible outcomes in 1 trial.

So P(4) = 4 / 52 = 1 / 13

Public domain image via Pixabay

What Is the Expectation of an Event?

Once a probability has been worked out, it's possible to get an estimate of how many events will likely happen in future trials. This is known as the expectation and is denoted by E.

If the event is A and the probability of A occurring is P(A), then for N trials, the expectation is:

E = P(A) N

For the simple example of a dice throw, the probability of getting a six is 1/6.
So in 60 trials, the expectation or number of expected 6's is:

E = 1/6 x 60 = 10

Remember, the expectation is not what will actually happen, but what is likely to happen. In 2 throws of a dice, the expectation of getting a 6 (not two sixes) is:

E = 1/6 x 2 = 1/3

However, as we all know, it's quite possible to get 2 sixes in a row, even though the probability is only 1 in 36 (see how this is worked out later). As N becomes larger, the actual number of events which happen will get closer to the expectation. So for example when flipping a coin, if the coin isn't biased, the number of heads will be closely equal to the number of tails. In reality a coin might be heavier on one side and similarly for a dice. Also they can be thrown differently each time. Both these factors can alter the actual probability of an event occurring compared to the mathematical probability.

Public domain image via Pixabay

Success or Failure?

The probability of an event can range from 0 to 1.

Remember

P(Event) = Number of ways the event can occur / The total number of possible outcomes

So for a dice throw

P(getting a number between 1 and 6 inclusive) = 6 / 6 = 1 (since there are 6 ways you can get "a" number between 1 and 6, and 6 possible outcomes)

P(getting a 7) = 0 / 6 = 0 (there are no ways the event 7 can occur in any of the 6 possible outcomes)

P(getting a 5) = 1 / 6 (only 1 way of getting a 5)

If there are 999 failures in 1000 samples

Empirical probability of failure = P(failure) = 999/1000 = 0.999

A probability of 0 means that an event will never happen.

A probability of 1 means that an event will definitely happen.

In a trial, if event A is a success, then failure is not A (not a success)

and:

P(A) + P(not A) = 1

Independent and Dependent Events

Events are independent when the occurrence of one event doesn't affect the probability of the other event.

So if a card is drawn from a pack, the probability of an ace is 4/52 = 1/13.

If the card is replaced, the probability of drawing an ace is still 1/13.

Two events are dependent if the occurrence of the first event affects the probability of occurrence of the second event.

If an ace is drawn from a pack and not replaced, there are only 3 aces left and 51 cards remaining, so the probability of drawing a second ace is 3/51.

For two events A and B where B depends on A, the probability of Event B occurring after A is denoted by P(B|A).

Mutually Exclusive and Non-Exclusive Events

Mutually exclusive events are events that cannot occur together. For instance in the throwing of a dice, a 5 and a 6 can't occur together. Another example is picking coloured sweets out of a jar. if an event is picking a red sweet, and another event is picking a blue sweet, if a blue sweet is picked, it can't also be a red sweet and vice versa.

Mutually non-exclusive events are events that can occur together. For instance when a card is drawn from a pack and the event is a black card or an ace card. If a black is drawn, this doesn't exclude it from being an ace. Similarly if an ace is drawn, this doesn't exclude it from being a black card.

Addition Law of Probability

Here is this law applied to different types of events.

Mutually Exclusive Events

For mutually exclusive (they can't occur simultaneously) events A and B

P(A or B) = P(A) + P(B)

Example 1

A sweet jar contains 20 red sweets, 8 green sweets and 10 blue sweets. If two sweets are pickets are picked out, what is the probability of picking a red or a blue sweet? (To keep things simple, the first sweet is returned so there are still 38 sweets to choose from when the second sweet is picked.)

The event of picking out a red sweet and picking out a blue sweet are mutually exclusive.

There are 38 sweets in total, so:

P(red) = 20/38 = 10/19

P(blue) = 10/38 = 5/19

P(red or blue) = P(red) + P(blue) = 10/19 + 5/19 = 15/19

© Eugene Brennan

Example 2

A dice is thrown and a card is drawn from a pack, what is the possibility of getting a 6 or an ace?

There is only one way of getting a 6, so:

P(getting a six) is 1/6

There are 52 cards in a pack and four ways of getting an ace. Also drawing an ace is an independent event to getting a 6 (the earlier event doesn't influence it).

P(getting an ace) is 4/52 = 1/13

P(getting a six or an ace) = P(getting a six) + P(getting an ace)

= 1/6 + 1/13 = (13 + 6)/78 = 19/78

Remember in these type of problems, how the question is phrased is important. So the question was to determine the probability of one event occurring "or" the other event occurring and so the addition law of probability is used.

Mutually Non-Exclusive Events

If two events A and B are mutually non-exclusive, then:

P(A or B) = P(A) + P(B) - P(A and B)

..or alternatively in set theory notation where "U" means the union of sets A and B and "∩" means the intersection of A and B:

P(A U B) = P(A) + P(B) - P(A ∩ B)

We effectively have to subtract the mutual events that are "double counted". You can think of the two probabilities as sets and we are removing the intersection of the sets and calculating the union of set A and set B.

The union of the two sets A and B is P(A U B) = P(A) + P(B) - P(A ∩ B). © Eugene Brennan

Example 3

A coin is flipped twice. Calculate the probability of getting a head in either of the two trials.

In this example we could get a head in one trial, in the second trial or in both trials.

Let H₁ be the event of a head in the first trial and H₂ be the event of a head in the second trial

P(H₁) = 1/2 and P(H₂) = 1/2 (there is only one way a head can occur in each trial and two possible outcomes)

and P(H₁ or H₂) = P(H₁) + P(H₂) - P(H₁ and H₂)

There are four possible outcomes, HH, HT, TH and TT and only one way heads can appear twice. So P(H₁ and H₂) = 1/4

So P(H₁ or H₂) = P(H₁) + P(H₂) - P(H₁ and H₂) = 1/2 + 1/2 - 1/4 = 3/4

For more information on mutually non-exclusive events, see this article:
Taylor, Courtney. "Probability of the Union of 3 or More Sets." ThoughtCo, Feb. 11, 2020, thoughtco.com/probability-union-of-three-sets-more-3126263.

Multiplication Law of Probability

For independent (the first trial doesn't affect the second trial) events A and B

P(A and B) = P(A) x P(B)

Example

A dice is thrown and a card drawn from a pack, what is the probability of getting a 5 and a spade card?

P(getting a 5) = number of ways of getting a 5 / total number of outcomes

= 1/6

There are 52 cards in the pack and 4 suits or groups of cards, aces, spades, clubs and diamonds. Each suit has 13 cards, so there are 13 ways of getting a spade.

So P(drawing a spade) = number of ways of getting a spade / total number of outcomes

= 13/52 = 1/4

So P(getting a 5 and drawing a spade)

= P(getting a 5) x P(drawing a spade) = 1/6 x 1/4 = 1/24

Again it's important to note that the word "and" was used in the question, so the multiplication law was used.

Recommended Books

Engineering Mathematics by K.A. Stroud, available on Amazon is an excellent math textbook for both engineering students and anyone with a general interest in mathematics. The material has been written for part 1 of BSc. Engineering Degrees and Higher National Diploma courses.

A wide range of topics are covered including matrices, vectors, complex numbers, calculus, calculus applications, differential equations and series. The text is written in the style of a personal tutor, guiding the reader through the content, posing questions and encouraging them to provide the answer. Personally, I've found it really easy to follow.

It also covers a more in-depth treatment of probability theory than what has been covered in this article plus a section on statistics.

This book basically makes learning mathematics fun!

Summary of Probability Rules

Rule 1: The probability of an event has a value between 0 and 1 inclusive.

0 ≤ P(A) ≤ 1

 

Rule 2: The sum of all probabilities adds up to 1.

If Ā is the compliment of A, or "not" A, i.e. event A not occurring, P(Ā) is the probability of A not occurring (or Ā occurring):

P(Ā) + P(A) = 1

 

Rule 3: The probability of an event not occurring is 1 minus the probability of it occurring.
This is also called the complement rule. It follows from rule 2:

It follows from rule 2 that the probability of an event not occurring is 1 - the probability of it occurring:

P(Ā) = 1 - P(A)

 

Rule 4: The probability of two independent events both occurring is the product of their probabilities.
This is also called the multiplication rule for independent events:

For two events A and B:

P(A and B) = P(A) x P(B)

 

Rule 5: The probability of either of two mutually exclusive events occurring is the sum of their probabilities.
This is also called the addition rule for mutually exclusive events:

For mutually exclusive events A and B

P(A or B) = P(A) + P(B)

 

Rule 6: The probability of either of two non-mutually exclusive events occurring is the sum of their probabilities minus the probability of both occurring.
This is also called the general addition rule:

For non-mutually exclusive events:

P(A or B) = P(A) + P(B) - P(A and B)

If A and B are independent, this becomes:

$$$$

Permutations and Combinations

To solve more difficult problems and derive an expression for the probability of a general binomial distribution, we need to understand the concept of permutations and combinations. I won't go into the mathematics of the derivation, but basically the expression is derived from the equation for working out combinations.

A Permutation Is an Arrangement

A permutation is a way of arranging a number of objects. So, for instance, if you have the letters A, B, and C then all the possible permutations are:

ABC, ACB, BAC, BCA, CAB, CBA

Note that BA is a different permutation to AB.

If you have n objects, there are n factorial number of ways of arranging them, written as n!

n! = n x (n-1) x (n-2) .... x 3 x 2 x 1

The reason for this is because for the first position, there are n choices, and for each of these choices, there are (n-1) choices for the second place (because 1 choice was used up for the first place), and for each of the choices in the first two places, (n-3) choices for the third place and so on.

In the example above, the 3 letters A, B, C could be arranged in 3! = 3 x 2 x 1 = 6 ways

In general, if n objects are selected r at a time then, the number of permutations is:

n! / (n-r)!

This is written as ⁿP_r

Example: 2 letters are chosen from the set of letters A, B, C, D. How many ways can the 2 letters be arranged?

There are 4 letters so n =4 and r = 2

ⁿP_r = ⁴P₂ = 4! / (4 - 2)! = 4! / 2! = 4 x 3 x 2 x 1 / 2 x 1 = 12

A Combination Is a Selection

A combination is a way of selecting objects from a set without regard to the order of the objects. So again if we have the letters A, B and C and select 3 letters from this set, there is only 1 way of doing this, i.e., select ABC.

If we select 2 letters at a time from ABC, all the possible selections are:

AB, AC, and BC

Remember, BA is the same selection as AB etc.

In general, if you have n objects in a set and make selections r at a time, the total possible number of selections is:

ⁿC_r = n! / ((n - r)! r!)

Example: 2 letters are chosen from the set ABCD. How many combinations are possible?

There are 4 letters so n = 4 and r = 2

ⁿC_r = ⁴C₂ = 4! / ( (4 - 2)! x 2!) = 4! / (2! x 2!)

= 4 x 3 x 2 x 1 / ( (2 x 1) x (2 x 1) ) = 6

General Binomial Distribution

In a trial, an event could be getting heads in a coin throw or a six in a throw of a dice.

If the occurrence of an event is defined as a, then

Let the probability of success be denoted by p

Let the probability of non-occurrence of the event or failure be denoted by q

p + q = 1

Let the number of successes be r

And n is the number of trials

Then

Example: What are the chances of getting 3 sixes in 10 throws of a dice?

There are 10 trials and 3 events of interest, i.e. successes so:

n = 10

r = 3

The probability of getting a 6 in a dice throw is 1/6, so:

p = 1/6

The probability of not getting a dice throw is:

q = 1 - p = 5/6

P(3 successes) = 10! / ((10 - 3)! 3!) x (5/6)^{(10 - 3)} x (1/6)³

= 10! / (7! x 3!) x (5/6)⁷ x (1/6)³

= 3628800 / (5040 x 6) x (78125 / 279936) x (1/216)

= 0.155

Note that this is the probability of getting exactly three sixes and not any more or less.

Winning the Lottery! How to Work out the Odds

We would all like to win the lottery, but the chances of winning are only slightly greater than 0. However "If you're not in, you can't win" and a slim chance is better than none at all!

Take, for example, the California State Lottery. A player must choose 5 numbers between 1 and 69 and 1 Powerball number between 1 and 26. So that is effectively a 5 number selection from 69 numbers and a 1 number selection from 1 to 26. To calculate the odds, we need to work out the number of combinations, not permutations, since it doesn't matter what way the numbers are arranged to win.

The number of combinations of r objects is ⁿC_r = n! / ((n - r)! r!)

n = 69

and

r = 5

and

ⁿC_r = ⁶⁹C₅ = 69! / ( (69 - 5)! 5!) = 69! / (64! 5!) = 11,238,513

So there are 11,238,513 possible ways of picking 5 numbers from a choice of 69 numbers.

Only 1 Powerball number is picked from 26 choices, so there are only 26 ways of doing this.

For every possible combination of 5 numbers from the 69, there are 26 possible Powerball numbers, so to get the total number of combinations, we multiply the two combinations.

So the total possible number of combinations = 11,238,513 x 26 = 292,201,338 or roughly 293 million and the probability of winning is 1 in 293 million.

References:

Stroud, K.A. (1970). Engineering Mathematics (3rd ed., 1987). Macmillan Education Ltd., London, England.

Suggested Reading

• How to Calculate Conditional Probabilities Using Bayes' Law
  Bayes' law is used to calculate conditional probabilities, such as the probability that A happens given that B happened. It has many applications but is often interpreted incorrectly.
• Bertrand's Paradox: A Problem in Probability Theory
  This is a look at Bertrand's Paradox, where a seemingly simple probability question yields some very different results.
• Borel’s Law of Probability
  The French mathematician Émile Borel proposed a law about the probability of events occurring.

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2016 Eugene Brennan

Eugene Brennan (author) from Ireland on May 08, 2019:

Not offhand. However I did a quick Google search for "games of chance probability books" and several were listed. Maybe you could check them out on Amazon and there might be customer reviews.

maurrice on May 08, 2019:

Thank you Eugene for this tutorial. Very Interesting! Do you recommend any book which goes into more detail, ideally exploring games of chance, sports books etc?

Eugene Brennan (author) from Ireland on April 30, 2019:

Hi maurrice,

The probability of the event is 1/6, so in 60 trials, the probability of that event is 1/6 + 1/6 + 1/6....... 60 times.

It's an "or" situation, so it's the probability of that event occurring in trial 1 or trial 2 or trial 3 etc up to trial 60.

So you add the probabilities.

If for instance you throw a dice and the event is getting a 6. Then if the question was "what is the expectation of getting a 6 in each trial", then you would multiply the probabilities because it's an "and" situation.

So it's the probability of a 6 in trial 1 and a 6 in trial 2 etc

= 1/6 x 1/6, 60 times = 1/6 ^ 60

In general,

Probability = number of ways event can occur / number of possible outcomes.

So taking the dice example again:

In two trials there's 12 ways you can get a 6:

1) 6 in the first trial and 6 other numbers in the second trial (6 possibilities)

2) 6 in the second trial and 6 other numbers in the first trial (6 possibilities)

The number of outcomes is 6 x 6 = 36

Since if you get 1 in the first trial, you can get 1 to 6 in the second trial

If you get 2 in the first trial, you can get 1 to 6 in the second trial and so on.

So probability = 12/36 = 1/3

So you get the same answer as by adding the probabilities because it’s an “or” situation

1/6 + 1/6 = 1/3

maurrice on April 29, 2019:

From the following section: What Is the Expectation of an Event?

Why is the answer calculated as 1/6 x 60?

Isn't it the same probability per trial, i.e.:

1st trial = 1/6 chance of getting any number

2nd trial = 1/6 chance of getting any number

and so on...

Therefore, why is it not calculated as (1/6)^60? What am I missing out/confusing, please?

Thanks.

Ekki on November 29, 2018:

Thank you so much for this article. It was most helpful. It answered questions that bothered me since the days in college!

Eugene Brennan (author) from Ireland on January 24, 2016:

Thanks Larry!

Larry Rankin from Oklahoma on January 24, 2016:

Wonderful insight into odds.

Eugene Brennan (author) from Ireland on January 21, 2016:

Thanks LM, I learned this stuff in school over 30 years ago, but it was refreshing to revisit it!

LM Gutierrez on January 21, 2016:

Thanks for sharing and reiterating the basic mathematics we learn in our early years of schooling! Actually, this topic is very useful in real life even if you engange in a field which does not deal much on numbers such as mine. I agree with Jodah, well-researched hub!

Eugene Brennan (author) from Ireland on January 18, 2016:

Thanks Jodah and well spotted! That's what I get for racing through the proof reading!

John Hansen from Australia (Gondwana Land) on January 18, 2016:

It's nice to know these equations and the odds of throwing certain numbers of dice, drawing a certain card etc. Very well researched hub , Eugene. However under the heading "Probability of an Event" it says; "There are two types of probability, empirical and empirical."(should the second one be "classical"?)

17th Century Dunlavin Headstones

A 17th century headstone in the old cemetery in Dunlavin village, Co. Wicklow. © Eugene Brennan

I visited the little park adjacent to the market house in Dunlavin on my Sunday cycle yesterday. I've cycled past it dozens of times over the last thirty years, but never actually went in. This was the location of a 17th-century church and graveyard, the graveyard now having been “recycled”, with the headstones stacked up against the back wall of the space. It’s a practice I’m not too fond of, and it's something that has occurred in several locations in Dublin—Cemeteries have been turned into parks, such as at the rear of St Mary’s Church (foundation stone laid in 1700) on Mary Street, and at St Kevin’s Church, Camden Row (behind the now-demolished Kevin Street College of Technology, my alma mater). It was a shock to discover on Street View that the college had gone, but it was an ugly building.
The cemetery in Dunlavin has a couple of old 17th-century headstones, something which isn’t commonplace, as the inscriptions on headstones older than the 18th century are normally eroded and illegible unless they’ve been sheltered from the elements.

Edit: Dunlavin local historian Chris Lawlor has kindly sent me a link to his thesis, The Establishment and Evolution of an Irish Village: The Case of Dunlavin, County Wicklow 1600 -1910, which includes some details about the cemetery on p. 44:

Map courtesy Tailte Éireann.

Another 17th century headstone in Dunlavin old cemetery. © Eugene Brennan 

The location of the cemetery in Dunlavin village. Image courtesy Tailte Éireann.

Chimney Stacks and Pots in Kilcullen

Chimney stacks, some topped with pots, on buildings along Kilcullen’s Upper Main Street. 

I've been reading the cover article from the April edition of The Bridge about the planned streetscape survey of Kilcullen's Upper Main Street. Zooming in on a c. 1900 image of the street, it appears that many of the chimney stacks were devoid of chimney pots—the same is true for Lower Main Street. A chimney pot serves several functions beyond mere aesthetics. It improves airflow by effectively raising the chimney's height without requiring the entire stack to be built taller. Since a chimney pot is narrower than the flue inside the stack (these chimney stacks were often wide and sometimes flue-less; in fact, children used to climb the stacks to sweep the chimneys), it creates a venturi effect, increasing airflow speed and improving suction.

Chimney pots became more common in the 18th century, with the peak of the fashion occurring in the 19th century.

Another feature of the chimney stacks is that many of them appear to have been constructed from brick, rather than rough stone. Over the intervening 130 years or so since the Lawrence Collection photo was taken, most of the stacks have been rendered with cement or lime mortar.

Images courtesy The National Library of Ireland.

Brick-constructed chimney stacks on buildings along Kilcullen's upper main street.

 

Lawrence Collection photo, c. 1900, of Kilcullen's Upper Main Street, 

 

How to Calculate Arc Length of a Circle, Segment and Sector Area

Circumference, diameter and radius. © Eugene Brennan

 

What You'll Learn

In this tutorial you'll learn about:

• names for different parts of a circle
• degrees and radians and how to convert between them
• chords, arcs and secants
• sine and cosine
• how to work out the length of an arc and chord
• how to calculate the area of sectors and segments
• the equation of a circle in the Cartesian coordinate system

What Is a Circle?

"A locus is a curve or other figure formed by all the points satisfying a particular equation."

A circle is a single-sided shape, but it can also be described as a locus of points where each point is equidistant (the same distance) from the centre.

Angle Formed by Two Rays Emanating From the Center of a Circle

An angle is formed when two lines or rays that are joined together at their endpoints, diverge or spread apart. Angles range from 0 to 360 degrees. We often "borrow" letters from the Greek alphabet to use in math and science. So for instance, we use the Greek letter "p" which is π (pi) and pronounced "pie" to represent the ratio of the circumference of a circle to the diameter. We also use the Greek letter θ (theta) and pronounced "the - ta", for representing angles.

An angle is formed by two rays diverging from the centre of a circle. This angle ranges from 0 to 360 degrees. © Eugene Brennan

360 degrees in a full circle. © Eugene Brennan

Parts of a Circle

• A sector is a portion of a circular disk enclosed by two rays and an arc.
• A segment is a portion of a circular disk enclosed by an arc and a chord.
• A semi-circle is a special case of a segment, formed when the chord equals the length of the diameter.

Arc, sector, segment, rays and chord. © Eugene Brennan

What Is Pi (π) ?

Pi represented by the Greek letter π is the ratio of the circumference to the diameter of a circle. It's a non-rational number which means that it can't be expressed as a fraction in the form a/b where a and b are integers.

Pi is equal to 3.1416 rounded to 4 decimal places.

What's the Length of the Circumference of a Circle?

If the diameter of a circle is D and the radius is R.

Then the circumference C = πD

But D = 2R

So in terms of the radius R

C = πD = 2πR

What's the Area of a Circle?

The area of a circle is A = πR ²

But R = D/2

So the area in terms of the radius R is

Scroll to Continue

A = πR ² = π (D/2)² = πD ²/4

What Are Degrees and Radians?

Angles are measured in degrees, but sometimes to make the mathematics simpler and elegant it's better to use radians which is another way of denoting an angle. A radian is the angle subtended by an arc of length equal to the radius of the circle. ( "Subtended" means produced by joining two lines from the end points of the arc to the center).

An arc of length R where R is the radius of a circle, corresponds to an angle of 1 radian.

So if the circumference of a circle is 2πR i.e 2π times R, the angle for a full circle will be 2π times 1 radian or 2π radians.

And 360 degrees = 2π radians.

How to Convert From Degrees to Radians

1. 360 degrees = 2π radians
2. Dividing both sides by 360 gives
3. 1 degree = 2π /360 radians
4. Then multiply both sides by θ
5. θ degrees = (2π/360) x θ = θ(π/180) radians
6. So to convert from degrees to radians, multiply by π/180

A radian is the angle subtended by an arc of length equal to the radius of a circle. © Eugene Brennan

How to Convert From Radians to Degrees

1. 2π radians = 360 degrees
2. Divide both sides by 2π giving
3. 1 radian = 360 / (2π) degrees
4. Multiply both sides by θ, so for an angle θ radians
5. θ radians = 360/(2π) x θ = (180/π)θ degrees
6. So to convert radians to degrees, multiply by 180/π

How to Find the Length of an Arc

You can work out the length of an arc by calculating what fraction the angle is of the 360 degrees for a full circle.

1. A full 360 degree angle has an associated arc length equal to the circumference C
2. So 360 degrees corresponds to an arc length C = 2πR
3. Divide by 360 to find the arc length for one degree:
4. 1 degree corresponds to an arc length 2πR/360
5. To find the arc length for an angle θ, multiply the result above by θ:
6. 1 x θ = θ corresponds to an arc length (2πR/360) x θ

So arc length s for an angle θ is:

s = (2πR/360) x θ = πRθ /180

The derivation is much simpler for radians:

By definition, 1 radian corresponds to an arc length R

So if the angle is θ radians, multiplying by θ gives:

Arc length s = R x θ = Rθ

Arc length is Rθ when θ is in radians. © Eugene Brennan

What Are Sine and Cosine?

A right-angled triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse and it is the longest side. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

In the diagram below, one of the angles is represented by the Greek letter θ.

The side a is known as the "opposite" side and side b is the "adjacent" side to the angle θ.

sine θ = length of opposite side / length of hypotenuse

cosine θ = length of adjacent side / length of hypotenuse

Sine and cosine apply to an angle, not necessarily an angle in a triangle, so it's possible to just have two lines meeting at a point and to evaluate sine or cos for that angle. However sine and cos are derived from the sides of an imaginary right angled triangle superimposed on the lines. In the second diagram below, you can imagine a right angled triangle superimposed on the purple triangle, from which the opposite and adjacent sides and hypotenuse can be determined.

Over the range 0 to 90 degrees, sine ranges from 0 to 1 and cos ranges from 1 to 0

Remember sine and cosine only depend on the angle, not the size of the triangle. So if the length a changes in the diagram below when the triangle changes in size, the hypotenuse c also changes in size, but the ratio of a to c remains constant.

Sine and cosine are sometimes abbreviated to sin and cos.

Sine and cosine of angles. © Eugene Brennan

How to Calculate the Area of a Sector of a Circle

The total area of a circle is πR ² corresponding to an angle of 2π radians for the full circle.

If the angle is θ, then this is θ/2π the fraction of the full angle for a circle.

So the area of the sector is this fraction multiplied by the total area of the circle

or

(θ/2π) x (πR ²) = θR ²/2 with θ in radians.

Area of a sector of a circle knowing the angle θ in radians. © Eugene Brennan

How to Calculate the Length of a Chord Produced by an Angle

The length of a chord can be calculated using the Cosine Rule.
For the triangle XYZ in the diagram below, the side opposite the angle θ is the chord with length c.

From the Cosine Rule:

c ² = R ² + R ² -2RRcos θ

Simplifying:

c ² = R ² + R ² -2R ²cos θor c ² = 2R ² (1 - cos θ)

But from the half-angle formula (1- cos θ)/2 = sin ² (θ/2) or (1- cos θ) = 2sin ² (θ/2)

Substituting gives:

c² = 2R ² (1 - cos θ) = 2R ²2sin ² (θ/2) = 4R ²sin ² (θ/2)

Taking square roots of both sides gives:

c = 2Rsin(θ/2) with θ in radians.

A simpler derivation arrived at by splitting the triangle XYZ into 2 equal triangles and using the sine relationship between the opposite and hypotenuse, is shown in the calculation of segment area below.

The length of a chord. © Eugene Brennan

How to Calculate the Area of a Segment of a Circle

To calculate the area of a segment bounded by a chord and arc subtended by an angle θ , first work out the area of the triangle, then subtract this from the area of the sector, giving the area of the segment. (see diagrams below)

The triangle with angle θ can be bisected giving two right angled triangles with angles θ/2.

sin(θ/2) = a/R

So a = Rsin(θ/2) (cord length c = 2a = 2Rsin(θ/2)

cos(θ/2) = b/R

So b = Rcos(θ/2)

The area of the triangle XYZ is half the base by the perpendicular height so if the base is the chord XY, half the base is a and the perpendicular height is b. So the area is:

ab

Substituting for a and b gives:

Rsin(θ/2)Rcos(θ/2)

= R ²sin(θ/2)cos(θ/2)

But the double angle formula states that sin(2θ) = 2sin(θ)cos(θ)

Substituting gives:

Area of the triangle XYZ = R ²sin(θ/2)cos(θ/2) = R ² ((1/2)sin θ) = (1/2)R ²sin θ

Also, the area of the sector is:

R ²(θ/2)

And the area of the segment is the difference between the area of the sector and the triangle, so subtracting gives:

Area of segment = R ²(θ/2) - (1/2)R ²sin θ

= (R ²/2)( θ - sin θ ) with θ in radians.

To calculate the area of the segment, first calculate the area of the triangle XYZ and then subtract it from the sector.. © Eugene Brennan

Area of a segment of a circle knowing the angle. © Eugene Brennan

Equation of a Circle in Standard Form

If the centre of a circle is located at the origin, we can take any point on the circumference and superimpose a right angled triangle with the hypotenuse joining this point to the centre.
Then from Pythagoras's theorem, the square on the hypotenuse equals the sum of the squares on the other two sides. If the radius of a circle is r then this is the hypotenuse of the right angled triangle so we can write the equation as:

x ² + y ² = r ²

This is the equation of a circle in standard form in Cartesian coordinates.

If the circle is centred at the point (a,b), the equation of the circle is:

(x - a)² + (y - b)² = r ²

The equation of a circle with a centre at the origin is r² = x² + y². © Eugene Brennan

Equation of a Circle in Parametric Form

Another way of representing the coordinates of a circle is in parametric form. This expresses the values for the x and y coordinates in terms of a parameter. The parameter is chosen as the angle between the x-axis and the line joining the point (x,y) to the origin. If this angle is θ, then:

x = cos θ

y = sin θ

for 0 < θ < 360°

Summary of Equations for a Circle

Table 1. Circle formulas. θ is in radians.

Quantity	Equation
Circumference	πD
Area	πR²
Arc Length	Rθ
Chord Length	2Rsin(θ/2)
Sector Area	R²θ/2
Segment Area	(R²/2) (θ - sin(θ))
Perpendicular distance from circle centre to chord	Rcos(θ/2)
Angle subtended by arc	arc length / (Rθ)
Angle subtended by chord	2arcsin(chord length / (2R))

Example

Here's a practical example of using trigonometry with arcs and chords. A curved wall is built in front of a building. The wall is a section of a circle. It's necessary to work out the distance from points on the curve to the wall of the building (distance "B"), knowing the radius of curvature R, chord length L, distance from chord to wall S and distance from centre line to point on curve A. See if you can determine how the equations were derived. Hint: Use Pythagoras's Theorem.

© Eugene Brennan

References

Harris, J., & Stöcker Horst. (1998). Handbook of Mathematical Formulas and Computational Science. Springer.

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2018 Eugene Brennan

 

Q&A From Readers 

George Dimitriadis from Templestowe on May 18, 2018:

Hi.

A good introduction to the basics of circle properties.

Diagrams are clear and informative.

Just a couple of points.

You have So C = πD = πR/2, which should be C = πD = 2πR

and A = πR^2 = π (D/2)2 = πD^2/2

should be A = πR^2 = π (D/2)2 = πD^2/4

Eugene Brennan (author) from Ireland on May 18, 2018:

Thanks George, I should have proof read before publishing, instead of beta testing on the readers !!

Larry Rankin from Oklahoma on May 19, 2018:

Very educational.

Troy Sartain on March 19, 2019:

How about a similar article for ellipses? Just a thought. Obviously, another level of complexity, even if not rotated.

Eugene Brennan (author) from Ireland on March 19, 2019:

Thanks Troy, I'll keep it in mind. Parabolas will probably come first though.

Mazin G A on April 01, 2019:

Hi,

How can I calculate the angle at the center of an arc knowing radius and center, start, and end points? I know how to do that if I have the length of the arc, but in my case I don't have it.

Eugene Brennan (author) from Ireland on April 05, 2019:

If you mean you know the coordinates of the start and end points of the chord, you can work out the length of the chord using Pythagoras's theorem. Then use the equation for length of a chord (2Rsin(θ/2) to find θ.

Lakshay on September 19, 2019:

Good efforts

darrell on April 06, 2020:

how do i calculate the length of a segment of a circle

Eugene Brennan (author) from Ireland on April 07, 2020:

If you mean the chord length, it's 2Rsin(θ/2).

See the derivation above.

Austen SMITH on April 28, 2020:

Hi I have a simple but frustrating problem- I want to build a regular curved wall a set distance from a straight wall - the centre of the circle /arc of the wall falls within the building.

I need to work out distance from the straight wall to measure, at regular intervals, to create the perfect curve starting and ending on the chord (2nd) forming the distance from the straight wall.(1st chord)

Hope you can help.

Eugene Brennan (author) from Ireland on April 30, 2020:

Hi Austen, I spent hours trying to figure this out using angles, but it turned out that since the chord length is known between two ends of the curved wall (is this correct?), it can easily be worked out using Pythagoras's Theorem. I've drawn it up as an example at the bottom of the article, hope it helps.

Suggestion, you could put the values into a spreadsheet to do the calculations.

Austen on May 31, 2020:

Many thanks for solving the curved wall issue.

I’ racked my brains back 45 plus used the existing formulae on your website to crack it- but as always when someone who “knows” tells you “how” - it becomes so clear you wonder how you couldn't see it before.

Thanks for relighting the knowledge thirst.

A

Eugene Brennan (author) from Ireland on May 31, 2020:

Thanks Austen.

I worked out D in the diagram above knowing R and L/2. In reality that's probably not necessary because you may already know the distance from the centre of the arc to the inside of the wall. Adding this to S gives you D.

Driving Slower to Reduce Fuel Consumption

Created by ChatGPT

In theory, according to Newton's first law, a body remains at rest, or in motion at a constant speed in a straight line, unless it is acted upon by a force. So if you kicked a ball in space and there were no planets or stars to influence the ball, it would go on forever.

How to Use a Multimeter to Measure Voltage, Current, and Resistance

© Eugene Brennan

 

What Is a Multimeter?

Multimeters are measurement instruments widely used by professionals in several fields including industrial maintenance and testing, research, appliance repair, and electrical installation. However, a digital multimeter (or DMM) is also an invaluable test instrument for home and DIY use.

Conversations With ChatGPT: Air Drag and Surface Topology

Created by Grok.

Eugene: Is it possible to make a material that has a huge amount of drag that could allow a human to float? And what topology of surface has the most drag, a plane? I know that drag is proportional to cross-sectional area and also that there is a coefficient that depends on the surface shape of an object.

Parabola Equations and Graphs, Directrix and Focus and How to Find Roots of Quadratic Equations

© Eugene Brennan

In this tutorial, you'll learn about a mathematical function called the parabola. We'll cover the definition of the parabola first and how it relates to the solid shape called the cone. Next, we'll explore different ways in which the equation of a parabola can be expressed.

Conversations With ChatGPT: Converting Garden Waste to Electricity

Created by ChatGPT

Eugene: Is it possible to convert the energy stored in garden waste directly into electricity?

Special Relativity and Angle Grinder Disks

© Eugene Brennan

The theory of special relativity was postulated by Einstein in his 1905 paper, "On the Electrodynamics of Moving Bodies". While things behave seemingly "normal" at speeds we experience in our daily lives, bizarre things happen at near-light speeds.

Where Does Water Come From in an Air Compressor Tank?

Graph of temperature versus max water storage capacity of air.

Anyone who owns an air compressor knows that one of the regular maintenance chores is to drain the tank to release water. This is essential to prevent eventual corrosion of the tank.

Rules of Logarithms and Exponents With Worked Examples and Problems

A graph of a log function. Krishnavedala , CC BY-SA 3.0 via Wikimedia Commons

An Introduction to Logarithms, Bases and Exponents

In this tutorial you'll learn about

• exponentiation
• bases
• logarithms to the base 10

Another Use for AI: Reading Tables From Images and Converting to HTML

Screenshot of a table. © Eugene Brennan

Artificial intelligence gets a lot of bad press, so I need to write a proper article at some stage about how beneficial I’ve found it to be for all sorts of tasks.

Summary of Ideal Gas Laws

Thanks to ChatGPT for creating the table to my design. There's a long discussion with ChatGPT about the gas law equations here, where I revise my limited knowledge from school. (which was basically just understanding Boyle's Law and Charles's Law)

Law / Equation	Form / Equation	Derived From	Notes / Conditions
Boyle’s Law	P V = constant	Ideal Gas Law P V = n R T, with T = constant	Isothermal: pressure inversely proportional to volume
Charles’s Law	V / T = constant or V ∝ T	Ideal Gas Law P V = n R T, with P = constant	Isobaric: volume directly proportional to temperature
Gay-Lussac’s Law	P / T = constant or P ∝ T	Ideal Gas Law P V = n R T, with V = constant	Isochoric: pressure directly proportional to temperature
Ideal Gas Law	P V = n R T	Fundamental definition of ideal gases	Connects pressure, volume, temperature, and moles
Combined Gas Law	P V / T = constant P1 V1 / T1 = P2 V2 / T2	Ideal Gas Law, general form combining Boyle, Charles, Gay-Lussac	Relates pressure, volume, and temperature for a fixed amount of gas
Adiabatic PV Relation	P Vγ = constant	First Law dU = -P dV + Ideal Gas Law, γ = Cp/Cv	No heat transfer; pressure rises faster than 1/V
Adiabatic T–V Relation	T Vγ-1 = constant	PV adiabatic relation + Ideal Gas Law	Temperature rises when volume decreases (or vice versa)
Adiabatic T–P Relation	T = constant × P(γ-1)/γ	From PV and T–V adiabatic relations	Temperature as a function of pressure
Adiabatic PV Relation (states)	P1 V1γ = P2 V2γ	PV adiabatic relation applied to two states	Useful for calculating compression/expansion between two points
Adiabatic T–V Relation (states)	T1 V1γ-1 = T2 V2γ-1	T–V adiabatic relation applied to two states	Temperature change for volume change between two states
Adiabatic T–P Relation (states)	T2 = T1 (P2/P1)(γ-1)/γ	T–P adiabatic relation applied to two states	Temperature change for pressure change between two states

Unorthodox Fire Starting Devices

A fire plunger. Image courtesy eBay

Two gadgets for those who need to light a fire when camping or otherwise and want to show off (although a cigarette or gas lighter is perfectly adequate):

How to Understand Calculus: Integration Rules and Examples

© Eugene Brennan

What Is Calculus?

Calculus is a study of rates of change of functions and accumulation of infinitesimally small quantities. It can be broadly divided into two branches: