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# Explorations with Pi Episode 22 of Geometry

## 36 episodes in this series

Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
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…
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…
Episode 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…
Episode 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…
Episode 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…
Episode 18 What Is the Sine of 1°?
So far, you've seen how to calculate the sine, cosine, and tangents of basic angles (0deg, 30deg, 45deg, 60deg, and 90deg). What about calculating them for other angles--without a calculator?…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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…
Episode 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.
Episode 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…
Episode 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…
Episode 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…
Episode 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.
Episode 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…
Episode 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…
Episode 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…