Introduction
The reason for this page is a memory from my childhood in 1961. As I watched Donald in Mathmagic Land (YouTube video) on Walt Disney’s Wonderful World of Color, I was hooked on mathematics. I especially was interested by the billiards segment that explaining how math figures were used in the game, and how the player works on his strategy based on the use of math.
Mathematical Billiards
Mathematical billiards is an idealisation of what we experience on a regular pool table. In mathematical billiards the ball bounces around according to the same rules as in ordinary billiards, but it has no mass, which means there is no friction. There also aren’t any pockets that can swallow the ball. This means that the ball will bounce infinitely many times on the sides of the billiard table and keep going forever.
One fascinating aspect of mathematical billiards is that it gives us a geometrical method to determine the least common multiple and the greatest common divisor of two natural numbers. Have a look at the Geogebra animation below (the play button is in the bottom left corner) and try to figure out how the construction works. If you would like to play the animation again, double click the refresh button in the top right corner. The two natural numbers are 40 and 15 in this case. The least common multiple of 40 and 15 equals 120, and the greatest common divisor is 5. [1]
“Pool is geometry, in its most challenging form.
The Twilight Zone: A Game of Pool
A science of precise angles and forces.”
Introduction to the Game
Here is some basic terminology dealing with a cue and a pool table (see also: pool table terminology): [5]
Here is some terminology related to a pool shot:
Diamond System
The diamonds on a pool table consist of nine diamonds and eight segments length-wise. Along with five diamonds and four segments width-wise. Yes, the pockets are included in the count. Having that two-to-one relationship is very important for geometric accuracy. Connecting the diamonds lead to learning the proper angles, this will helps you understand how the cue ball will bounce off the cushions in order to produce the desired impact. [2]
Bank Shot
One way the diamonds can be helpful when playing is to aid your bank shot. Bank shots became an integral part of the game after Charles Goodyear invented vulcanized rubber, making these shots consistent and reliable. Today, bank shots remain a sought-after skill to master.
To use the diamonds for your bank shot, find the one that will make the object ball, the ball you want to hit, bounce off the rail and drop into the desired pocket. You also have to ensure that the cue ball will not interfere with the object ball’s path after impact when you’re planning your bank shot.
For example, if the cue ball is at the side pocket, the object ball is on the corner pocket and the straight path along the rail is blocked, you may want to bank. Here’s how you might use the diamond system to make the bank:
- First, remember that the angle that the ball enters the rail is the angle that it will leave, known as mirrored angles.
- Use the numbers from the beginning for the two rails involved.
- Since the side pocket is four diamonds away from the corner, divide by two and hit the opposite rail on the second diamond, halfway between the corner and side pockets. By using the suggested diamond, the distance is cut in half, mirrored angles are created and the ball travels correctly.
This is just one example of how you might use the diamond system for banking. The possibilities are endless depending on the layout of the cue ball, the object ball and any other balls blocking the path. [3]
The Tangent Line
The Tangent Line is one of the most important things to know regarding position play. Simply defined, the tangent line is the natural direction a sliding cue ball takes after contacting an object ball. If a cue ball is rolling when it contacts an object ball (Topspin), or spinning backwards (Draw), when it contacts an object ball, the cue ball will roll forward or behind the tangent line.
The tangent line is perpendicular to a line drawn through the object ball and cue ball at the point of contact. This is a 90 degree angle from the path of the object ball to the pocket. On all shots the cue ball will travel this path unless it is shot with follow or draw.
Depending on the circumstances you may want to have the cue ball follow this line. If that is the case you need to have the cue ball sliding, not rolling or spinning when it makes contact with the object ball. Depending on the distance between the object and cue ball, you have to adjust your contact point on the cue ball or the hardness of the hit. This takes a lot of trial and error to find the best combination of speed and cue tip location.
In some cases you simply need to hit harder to keep the cue ball sliding over a longer distance. In some shots you need to hit lower on the cue ball so it starts out spinning backwards but has stopped spinning at the moment of contact with the cue ball.
Identifying and understanding the tangent line on each and every shot will become a guide to your controlling the cue ball and becoming successful at position play. You can use forward roll or topspin to have the cue ball roll forward of the tangent line, or use a draw stroke to have the cue ball move behind the tangent line.
Learning these techniques and using the tangent line will improve your success at position play. You can adjust your position far more effectively with topspin or draw once you know where that tangent line is. Often this skill is very important to help you avoid scratching or breaking out a cluster of balls. It’s all about The tangent line that you want to follow, go over or stay behind with the cue ball. This is a decision to make on every shot. [4]
The tangent line is is the natural line that the cue ball will travel down after impact with the object ball when hit with stun (no top or bottom spin). This is the natural line that the cue ball wants to travel down and spin is the only thing that will take it off of that line. The tangent line is drawn at 90 degrees to the line that the object ball will travel, but starts from the center of the cue ball at impact with the object ball. This is the single most important concept of cue ball control. [6]
Training Cue Balls
It would seem that if you were able to aim perfectly on every shot, you would pocket every ball. However, when a player misses a shot, it is rarely their aim that has let them down. The majority of even average players know how to aim a majority of shots without any aiming system. The reason for most misses is poor delivery of the cue to and through the cue ball. No aiming system in the world will help the player who can’t deliver a precise and accurate stroke. [7]
Great cue ball control is derived from your knowledge and experience on how to send the cue ball on a specific path by choosing the right spin, speed, and angle, and then executing that shot with precise cuing. This section will focus on giving you the knowledge you need to control the cue ball’s path. This includes the three factors that determine the path of the cue ball after striking the object ball; tangent line, spin, and speed. [6]
To train and become better at pool and cue sports, players need a training cue ball that helps them know where to hit the cue ball to pocket the object ball. The billiard training ball allows players to find the right areas to hit on both the object ball and the cue ball for a cut shot. By using a geometric design, the practice cue ball and object balls allow players to easily see the spots that need to be hit to sink a pool or snooker ball into the pocket.
For example, the Aramith Jim Rempe Training Cue Ball (YouTube video) is the product of a collaborated effort between the renowned billiard brand Aramith and famed professional pocket billiards player James Rempe. The regulation-sized ball features various marking to assist players with practicing their topspin, sidespin, backspin, and other crucial forms of cue ball control. It’s also paired with a 56-page instruction manual that helps guide the reader through the process of using this tool correctly to improve their skills. Of all the training cue balls you’ll see listed on online sources, this is by the most highly recommended. However, its quality materials and reputation come with a hefty price tag you’ll want to consider heavily prior to purchase.
Problems
What is a page about pool table geometry without problems to solve! Here are some web pages to reinforce the geometric aspect of playing pool.
Finkel, Dan. 2023. “The Award-Winning Lesson Plan – The Billiard Ball Problem – Math For Love”. Math For Love. https://mathforlove.com/2023/02/the-award-winning-lesson-plan-the-billiard-ball-problem/.
My students became captivated by the question of predicting where a ball will bounce on a rectangular table of different dimensions. At first it seemed straightforward, and reducing the question to smaller tables led them to believe they had the answer. But some student would always remain skeptical and find counterexamples that shot down their peers’ conjectures. The work of extending theories, demonstrating counterexamples, and refining conjectures was suddenly the central work of my students. They behaved, in short, like mathematicians. As a result, the Billiard Ball Problem has become one of my favorite lessons. I’ve seen it create a kind of intellectual aliveness in students. I’ve also seen it create a sense of ownership, both about the experience and arguments from the activity, and about mathematics in general.
Furman, Stephanie and Hema Gopalakrishnan. “Pool Table Geometry”. 2023. Darien Public Schools, Sacred Heart University. http://sigmaa.maa.org/mcst/PosterActivitySessions/documents/BilliardsPoster.pdf.
“Geometry Word Problem: A Perfect Pool Shot (Video) | Khan Academy”. 2023. Khan Academy. https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-geo-similar-and-congruent-triangles-modeling/v/triangle-similarity-in-pool.
⭐ “Pool Table Problem – Mathigon”. 2023. Mathigon. https://mathigon.org/task/pool-table-problem.
“Section 16.3: Geometric Problem Solving Using Transformations”. 2023. math.ksu.edu. https://www.math.ksu.edu/~mbscott/math320/grp-16-3sol.pdf.
References
[1] “Arithmetic Billiards”. 2023. Plus Maths. https://plus.maths.org/content/arithmetic-billiards-0.
[2] McDonald, Jesse. 2017. “Diamond System And Related Terminology – Golden West Billiards”. Golden West Billiards. https://billiardmfg.com/diamond-system-and-related-terminology/.
[3] “The Pool Diamond System Explained”. 2023. hbhomefurnishings.com. https://www.hbhomefurnishings.com/post/the-pool-diamond-system-explained.
[4] “The Tangent Line”. 2022. Billiards.Com, Inc . https://www.billiards.com/blogs/articles/the-tangent-line.
[5] “Introduction To The Game – Billiards And Pool Principles, Techniques, Resources”. 2023. Billiards And Pool Principles, Techniques, Resources. https://billiards.colostate.edu/tutorial/intro/.
[6] “How To Control The Cue Ball In Pool”. 2022. World of Pool and Billiards. https://www.worldofpoolandbilliards.com/post/cue-ball-control.
[7] “How To Aim In Pool”. 2022. World Of Pool And Billiards. https://www.worldofpoolandbilliards.com/post/how-to-aim-in-pool.
Dr. Dave Pool Info
Great reference with instruction, resources, rules, images and videos to explain pool!
“How to Play Pool.” 2025. Dr. Dave Pool Info. April 18. https://drdavepoolinfo.com/.
“Answers to All Questions.” 2025. Dr. Dave Pool Info. May 2. https://drdavepoolinfo.com//faq/.
“Templates and Diagrams.” 2025. Dr. Dave Pool Info. May 17. https://drdavepoolinfo.com/resources/templates/.
Additional Reading
“11 Billiard Pool Tips Every Beginner Needs To Learn | Supreme Billiards”. 2017. Supreme Billiards | Great Resource For Billiard Shooting Pool Tips, Tricks, Reviews, And More. https://supremebilliards.com/11-billiard-pool-tips-every-beginner-needs-know/.
“90/30 Degree Rule Follow-up – Part I”. 2023. billiards.colostate.edu. https://billiards.colostate.edu/bd_articles/2005/feb05.pdf.
“Beginner’s Guide to Bank Shots – Emerald Spas & Billiards”. 2017. Emerald Spas & Billiards. https://emeraldgr.com/beginners-guide-bank-shots/.
⭐ “Blog | World Of Pool And Billiards”. 2023. World Of Pool And Billiards. https://www.worldofpoolandbilliards.com/blog.
⭐ “Bringing Geometry to the Billiard Table”. 2023. CueAndMe. https://cueandme.com/angle-detective/theory/geometry-to-the-table/.
Everything on the CueAndMe.com website is being freely shared with the sincere hope that those who find value in it will return to the donate page above and donate what they feel is appropriate.
THEORY
APPLICATION
TRAINING
ANGLE DETECTIVE VIDEO GAME
“Carom Billiards Instructions”. 2023. mathorama.com. https://www.mathorama.com/geom/lessons/pool_howto.html.
“Cue Ball Direction For All Types Of Shots – Billiards And Pool Principles, Techniques, Resources”. 2023. Billiards And Pool Principles, Techniques, Resources. https://billiards.colostate.edu/faq/cue-ball-control/where-CB-goes/.
Di Maria, Giovanni. “The Laws of Physics in Billiards”. 2021. EETimes. https://www.eetimes.eu/the-laws-of-physics-in-billiards/.
⭐ “Diamond System – THE Universal System”. 2023. billardcf.cluster010.ovh.net. http://billardcf.cluster010.ovh.net/wp-content/uploads/2011/10/03-DiamondSystem-En.pdf.
“Explanation of the Diamond Systems”. 2023. tonnycarlsen.com. http://tonnycarlsen.com/wp-content/uploads/TCBS-Setup.pdf.
“How To Estimate A Cut Angle – Billiards And Pool Principles, Techniques, Resources”. 2023. Billiards And Pool Principles, Techniques, Resources. https://billiards.colostate.edu/faq/cut/estimating-angle/.
“How To Play Pool – Billiards And Pool Principles, Techniques, Resources”. 2023. Billiards And Pool Principles, Techniques, Resources. https://billiards.colostate.edu/.
“How to Play Pool From the Basics to Advanced Strategies”. 2023. azpl.net. https://www.azpl.net/How%20to%20Play%20Pool%20WithTOC.pdf.
“How to Play Pool Like a Mathematician”. 2023. wikiHow. https://www.wikihow.com/Play-Pool-Like-a-Mathematician.
Billiard balls collide with nearly perfect elasticity. This means that the kinetic energy in their motion is almost completely preserved, and very little of it dissipates into heat or other energy sinks. This makes pool and billiards a great sport to analyze mathematically. If you have perfect control over how you strike the cue ball and where to aim it, you can always predict what will happen.
“How to Use the Dots on a Pool Table: Easy Tricks to Become a Pro”. 2023. recroompick.com. https://recroompick.com/how-to-use-the-dots-on-a-pool-table/.
“Introduction To The Pool Diamond System”. 2023. Liberty GAMES. https://www.libertygames.co.uk/pool-diamond-system/.
The majority of American billiard tables have them, most professional players swear by them and beginners wonder what their purpose is. The Diamond System is used on American pool tables to improve accuracy and overall performance in a game of billiards. And with a simple bit of practice (and a touch of geometry!), it can be used by you too. The following guide applies to 9 foot tables.
Take a look at an American billiards table. On nearly every table you’ll notice little marks inlaid into the frame of the rails, often a table manufacturer will use diamonds as the markers It is these diamonds which can assist you in choosing the proper angles for kick shots and banks and ultimately act as your aiming system. The diamonds form invisible lines on a table, which go from diamond to diamond; from left to right. The general principle is if you hit the cue ball perfectly straight at one diamond, it will travel completely straight back towards the diamonds at the opposite end of the table and when used properly, you can calculate or visualise the precise impact point to send the cue ball (or object ball) to any other spot on the pool table. So if you hit from ten degrees, either way, it should rebound ten degrees out in the opposite direction. Simple! However, if you find yourself playing a situation where a diamond isn’t in the right spot, this is when you have to start using your imagination and envision “imaginary diamonds” in the appropriate spot, along with the corresponding imaginary lines.
Littleton. “USE MATH TO IMPROVE POOL AND POOL TO IMPROVE MATH”. 2018. Mathnasium LLC. https://www.mathnasium.com/blog/use-math-to-improve-pool-and-pool-to-improve-math-872090749,
“Notes on the Diamond System for playing Three-Cushion Billiards”. 2023. qedcat.com. https://www.qedcat.com/misc/billiards.pdf.
“Pattern Play Principles”. 2023. billiards.colostate.edu. https://billiards.colostate.edu/bd_articles/2021/feb21.pdf.
Pearson, Dave. 2022. “What Are Pool Bank Shots? 5 Pool Bank Shot Tips For Beginners”. Pearson Cues. https://pearsoncues.com/blog/pool-bank-shots/.
“Rack (Billiards) – Wikipedia”. 2010. en.wikipedia.org. https://en.wikipedia.org/wiki/Rack_(billiards).
⭐ “The Fundamentals | World Of Pool And Billiards”. 2023. World Of Pool And Billiards. https://www.worldofpoolandbilliards.com/the-fundamentals.
“What Are The Dots On A Pool Table For? – Billiard Beast”. 2023. billiardbeast.com. https://billiardbeast.com/what-are-the-dots-on-a-pool-table-for/.
The dots or diamonds on a pool table are there for two main reasons – to divide the table into distinct sections and to provide visual reference points for the geometry of certain shots. They are particularly useful for improving accuracy on bank shots and kick shots. However, the effective use of these markers depends on a given player’s understanding of how they work.
““THE” Diamond System – Corner Five – Part 1”. 2023. mypoolblog.com. http://www.mypoolblog.com/uploads/1/6/7/6/16764298/k6_-_corner_five_part_1.pdf.
Videos
Some knowledge with regard to the geometry of a snooker table can enhance your play. Here Barry demonstrates a bit of table geometry which could help you determine the outcome of your shot and help to place you in a better position against your opponent. Barry also refers back to yesteryear and how knowledge gained from billiards helped snooker legend Joe Davis become such a great player.
Dr. Dave and Tom Ross describe, illustrate, and demonstrate how to use the 30-degree-rule natural angle to plan various types of shots.
Sorry there’s no audio. In this construction we are looking at a billiards table and would like to have Ball C carom (i.e., a shot in pool in which an object ball strikes another ball before falling into a pocket) off of Wall X then Wall Y and hit Ball B. The steps to completing the construction are listed below:
1) Sketch an approximate path the ball will take.
2) Reflect B over Line (Wall) Y, label the new point B’.
3) Reflect B’ over Line (Wall) X, label the new point B”.
4) Draw a projection line from Point (Ball) C to point B”. (Darken the actual path of the ball)
5) Draw a projection line from the intersection of Line (Wall) X and the projection line from Step 4 to B’. (Darken the actual path of the ball)
6) Connect the intersection of Line (Wall) Y and the projection line from Step 5 to Point (Ball) B.
Dr. Dave demonstrates the 10 most useful systems for aiming bank and kick shots. Some are “diamond systems” that use the markings on the rails as references. Others are completely visual and require no diamonds or numbers. All of them are useful to know in different situations, especially if you do not yet have perfect intuition judging bank and kick shots.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
The featured image on this page is from the XARTESR website.
Mathematical Mysteries 