Geometry: An Interactive Journey to Mastery

Rated 5 out of 5 by from This is absolutely brilliant! This lecture is so great that I am having a hard time of stopping watching it. The professor explained the concepts so well and so creative. Love it! When the professor mentions some mathematician's names, it will be even better if the names can appear on the screen.
Date published: 2021-03-30
Rated 5 out of 5 by from Excellent Presentation I am a retired engineer who studied Geometry in the late sixties. Professor Tanton worked through this subject in a most creative and logical way. He sort of followed Euclid's Elements and then expanded on them. His detailed explanation of Astronomy and semi circles leading to right angles and Trigonometric functions was the best I have ever heard. He made this course logical and enjoyable. I would sincerely recommend this course to any high school student or anyone who really wants to understand Geometry.
Date published: 2021-03-16
Rated 5 out of 5 by from You didn't know that Geometery could be this cool! Coming from a high school geometry nerd, this was a great course. Fascinating and creative new ways to look at "plain" geometry.
Date published: 2021-03-07
Rated 5 out of 5 by from Great Course Prof T gets the old wheels in my head moving, and to quote a lady "in a most enjoyable way".
Date published: 2020-12-29
Rated 5 out of 5 by from wonderful I am a retired physician who enjoyed mathematics as a youth. This course was a delightful refresher for the principles and concepts in geometry. Professor Tanton is one of the most enthusiastic and exciting teachers I have ever had. This is clearly a course in which a text and additional reading/study is required but not necessary. Highly recommend this course.
Date published: 2020-12-04
Rated 4 out of 5 by from Triangle with more than 180 degrees internal angle Chapter on parallelism. He drew a triangle with angles of 105, 75 and x and then said it was not a triangle. Sure looked like one to me. Apparently the two given angles in the drawing were not 105 and 75. Maybe a note could be superimposed on that section to clarify that. Otherwise I really enjoyed the review of geometry I had fun with in high school 60myears ago.
Date published: 2020-10-10
Rated 5 out of 5 by from great review for an old mind I am retired teacher and military. I had geometry in high school 1965, and wanted a review, to keep my 73 year old mind alert. It is a great course, which I wish was available back in '65'. (sounds like pioneer days, saying that) It got the old cogs working. I thoroughly enjoyed it
Date published: 2020-08-04
Rated 5 out of 5 by from Geom The instructor is great he uses tools to help explain the topic wish I had this course when I went through high school geom he definitely gave me great ideas totally different when I learned it high school gym teacher taught me it
Date published: 2020-07-21
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Geometry: An Interactive Journey to Mastery
Course Trailer
Geometry-Ancient Ropes and Modern Phones
1: Geometry-Ancient Ropes and Modern Phones

Explore the origins of one of the oldest branches of mathematics. See how geometry not only deals with practical concerns such as mapping, navigation, architecture, and engineering, but also offers an intellectual journey in its own right-inviting big, deep questions.

33 min
Beginnings-Jargon and Undefined Terms
2: Beginnings-Jargon and Undefined Terms

Lay the basic building blocks of geometry by examining what we mean by the terms point, line, angle, plane, straight, and flat. Then learn the postulates or axioms for how those building blocks interact. Finally, work through your first proof-the vertical angle theorem.

28 min
Angles and Pencil-Turning Mysteries
3: Angles and Pencil-Turning Mysteries

Using nothing more than an ordinary pencil, see how three angles in a triangle can add up to 180 degrees. Then compare how the experience of turning a pencil on a flat triangle differs from walking in a triangular shape on the surface of a sphere. With this exercise, Professor Tanton introduces you to the difference between flat and spherical geometry

28 min
Understanding Polygons
4: Understanding Polygons

Shapes with straight lines (called polygons) are all around you, from the pattern on your bathroom floor to the structure of everyday objects. But although we may have an intuitive understanding of what these shapes are, how do we define them mathematically? What are their properties? Find out the answers to these questions and more.

31 min
The Pythagorean Theorem
5: The Pythagorean Theorem

We commonly define the Pythagorean theorem using the formula a2 + b2 = c2. But Pythagoras himself would have been confused by that. Explore how this famous theorem can be explained using common geometric shapes (no fancy algebra required), and how it's a critical foundation for the rest of geometry.

29 min
Distance, Midpoints, and Folding Ties
6: Distance, Midpoints, and Folding Ties

Learn how watching a fly on his ceiling inspired the mathematician Rene Descartes to link geometry and algebra. Find out how this powerful connection allows us to use algebra to calculate distances, midpoints, and more.

29 min
The Nature of Parallelism
7: The Nature of Parallelism

Examine how our usual definition of parallelism is impossible to check. Use the fundamental assumptions from the previous lectures to follow in Euclid's footsteps and create an alternative way of checking if lines are parallel. See how, using this result, it's possible to compute the circumference of the Earth just by using shadows!

35 min
Proofs and Proof Writing
8: Proofs and Proof Writing

The beauty of geometry is that each result logically builds on the others. Mathematicians demonstrate this chain of deduction using proofs. Learn this step-by-step process of logic and see how to construct your own proofs.

29 min
Similarity and Congruence
9: Similarity and Congruence

Define what it means for polygons to be "similar" or "congruent" by thinking about photocopies. Then use that to prove the third key assumption of geometry-the side-angle-side postulate-which lets you verify when triangles are similar. Thales of Ionia used this principle in 600 B.C.E. to impress the Egyptians by calculating the height of the pyramids.

34 min
Practical Applications of Similarity
10: Practical Applications of Similarity

Build on the side-angle-side postulate and derive other ways of testing whether triangles are similar or congruent. Also dive into several practical applications, including a trick botanists use for estimating the heights of trees and a way to measure the width of a river using only a baseball cap.

31 min
Making Use of Linear Equations
11: Making Use of Linear Equations

Delve deeper into the connections between algebra and geometry by looking at lines and their equations. Use the three basic assumptions from previous lectures to prove that parallel lines have the same slope and to calculate the shortest distance between a point and a line.

29 min
Equidistance-A Focus on Distance
12: Equidistance-A Focus on Distance

You've learned how to find the midpoint between two points. But what if you have three points? Or four points? Explore the concept of equidistance and how it reveals even more about the properties of triangles and other shapes.

33 min
A Return to Parallelism
13: A Return to Parallelism

Continue your study of parallelism by exploring the properties of transversals (lines that intersect two other lines). Prove how corresponding angles are congruent, and see how this fact ties into a particular type of polygon: trapezoids.

31 min
Exploring Special Quadrilaterals
14: Exploring Special Quadrilaterals

Classify all different types of four-sided polygons (called quadrilaterals) and learn the surprising characteristics about the diagonals and interior angles of rectangles, rhombuses, trapezoids, and more. Also see how real-life objects-like ironing boards-exhibit these geometric characteristics.

30 min
The Classification of Triangles
15: The Classification of Triangles

Continue the work of classification with triangles. Find out what mathematicians mean when they use words like scalene, isosceles, equilateral, acute, right, and obtuse. Then, learn how to use the Pythagorean theorem to determine the type of triangle (even if you don't know the measurements of the angles).

30 min
Circle-ometry-On Circular Motion
16: Circle-ometry-On Circular Motion

How can you figure out the "height" of the sun in the sky without being able to measure it directly with a ruler? Follow the path of ancient Indian scholars to answer this question using "angle of elevation" and a branch of geometry called trigonometry. You learn the basic trig identities (sine, cosine, and tangent) and how physicists use them to describe circular motion.

32 min
Trigonometry through Right Triangles
17: Trigonometry through Right Triangles

The trig identities you explored in the last lecture go beyond circles. Learn how to define all of them just using triangles (expressed in the famous acronym SOHCAHTOA). Then, uncover how trigonometry is practically applied by architects and engineers to measure the heights of buildings.

28 min
What Is the Sine of 1?
18: What Is the Sine of 1?

So far, you've seen how to calculate the sine, cosine, and tangents of basic angles (0°, 30°, 45°, 60°, and 90°). What about calculating them for other angles-without a calculator? You'll use the Pythagorean theorem to come up with formulas for sums and differences of the trig identities, which then allow you to calculate them for other angles.

32 min
The Geometry of a Circle
19: The Geometry of a Circle

Explore the world of circles! Learn the definition of a circle as well as what mathematicians mean when they say things like radius, chord, diameter, secant, tangent, and arc. See how these interact, and use that knowledge to prove the inscribed angle theorem and Thales' theorem.

29 min
The Equation of a Circle
20: The Equation of a Circle

In your study of lines, you used the combination of geometry and algebra to determine all kinds of interesting properties and characteristics. Now, you'll do the same for circles, including deriving the algebraic equation for a circle.

33 min
Understanding Area
21: Understanding Area

What do we mean when we say "area"? Explore how its definition isn't quite so straightforward. Then, work out the formula for the area of a triangle and see how to use that formula to derive the area of any other polygon.

28 min
Explorations with Pi
22: Explorations with Pi

We say that pi is 3.14159 ... but what is pi really? Why does it matter? And what does it have to do with the area of a circle? Explore the answer to these questions and more-including how to define pi for shapes other than circles (such as squares).

31 min
Three-Dimensional Geometry-Solids
23: Three-Dimensional Geometry-Solids

So far, you've figured out all kinds of fun properties with two-dimensional shapes. But what if you go up to three dimensions? In this lecture, you classify common 3-D shapes such as cones and cylinders, and learn some surprising definitions. Finally, you study the properties (like volume) of these shapes.

32 min
Introduction to Scale
24: Introduction to Scale

If you double the side-lengths of a shape, what happens to its area? If the shape is three-dimensional, what happens to its volume? In this lecture, you explore the concept of scale. You use this idea to re-derive one of our fundamental assumptions of geometry, the Pythagorean theorem, using the areas of any shape drawn on the edges of the right triangle-not just squares.

30 min
Playing with Geometric Probability
25: Playing with Geometric Probability

Unite geometry with the world of probability theory. See how connecting these seemingly unrelated fields offers new ways of solving questions of probability-including figuring out the likelihood of having a short wait for the bus at the bus stop.

30 min
Exploring Geometric Constructions
26: Exploring Geometric Constructions

Let's say you don't have a marked ruler to measure lengths or a protractor to measure angles. Can you still draw the basic geometric shapes? Explore how the ancient Greeks were able to construct angles and basic geometric shapes using no more than a straight edge for marking lines and a compass for drawing circles.

29 min
The Reflection Principle
27: The Reflection Principle

If you're playing squash and hit the ball against the wall, at what angle will it bounce back? If you're playing pool and want to play a trick shot against the side edge, how do you need to hit the ball? Play with these questions and more through an exploration of the reflection principle.

31 min
Tilings, Platonic Solids, and Theorems
28: Tilings, Platonic Solids, and Theorems

You've seen geometric tiling patterns on your bathroom floor and in the works of great artists. But what would happen if you made repeating patterns in 3-D space? In this lecture, discover the five platonic solids! Also, become an artist and create your own beautiful patterns-even using more than one type of shape.

32 min
Folding and Conics
29: Folding and Conics

Use paper-folding to unveil sets of curves: parabolas, ellipses, and hyperbolas. Study their special properties and see how these curves have applications across physics, astronomy, and mechanical engineering.

29 min
The Mathematics of Symmetry
30: The Mathematics of Symmetry

Human aesthetics seem to be drawn to symmetry. Explore this idea mathematically through the study of mappings, translations, dilations, and rotations-and see how symmetry is applied in modern-day examples such as cell phones.

28 min
The Mathematics of Fractals
31: The Mathematics of Fractals

Explore the beautiful and mysterious world of fractals. Learn what they are and how to create them. Examine famous examples such as Sierpinski's Triangle and the Koch Snowflake. Then, uncover how fractals appear in nature-from the structure of sea sponges to the walls of our small intestines.

30 min
Dido's Problem
32: Dido's Problem

If you have a fixed-length string, what shape can you create with that string to give you the biggest area? Uncover the answer to this question using the legendary story of Dido and the founding of the city of Carthage.

31 min
The Geometry of Braids-Curious Applications
33: The Geometry of Braids-Curious Applications

Wander through the crazy, counterintuitive world of rotations. Use a teacup and string to explore how the mathematics of geometry can describe an interesting result in quantum mechanics.

29 min
The Geometry of Figurate Numbers
34: The Geometry of Figurate Numbers

Ponder another surprising appearance of geometry-the mathematics of numbers and number theory. Look into the properties of square and triangular numbers, and use geometry to do some fancy arithmetic without a calculator.

32 min
Complex Numbers in Geometry
35: Complex Numbers in Geometry

In lecture 6, you saw how 17th-century mathematician Rene Descartes united geometry and algebra with the invention of the coordinate plane. Now go a step further and explore the power and surprises that come from using the complex number plane. Examine how using complex numbers can help solve several tricky geometry problems.

32 min
Bending the Axioms-New Geometries
36: Bending the Axioms-New Geometries

Wrap up the course by looking at several fun and different ways of reimagining geometry. Explore the counterintuitive behaviors of shapes, angles, and lines in spherical geometry, hyperbolic geometry, finite geometry, and even taxi-cab geometry. See how the world of geometry is never a closed-book experience.

32 min
James Tanton

Our complex society demands not only mastery of quantitative skills, but also the confidence to ask new questions, to explore, wonder, flail, to rely on ones wits, and to innovate. Let's teach joyous and successful thinking.


Princeton University


The Mathematical Association of America

About James Tanton

Dr. James Tanton is the Mathematician in Residence at The Mathematical Association of America (MAA). He earned a Ph.D. in Mathematics from Princeton University. A former high school teacher at St. Mark's School in Southborough and a lifelong educator, he is the recipient of the Beckenbach Book Prize from the MAA, the George Howell Kidder Faculty Prize from St. Mark's School, and a Raytheon Math Hero Award for excellence in math teaching. Professor Tanton is the author of a number of books on mathematics including Solve This: Math Activities for Students and Clubs, The Encyclopedia of Mathematics, and Mathematics Galore! Professor Tanton founded the St. Mark's Institute of Mathematics, an outreach program promoting joyful and effective mathematics education. He also conducts the professional development program for Math for America in Washington, D.C.

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