Summary
| Subject keyword(s) | Area, Area of circles, Circles, Famous numbers, Geometry, Mathematics, Mathematics history, Measurement, Number and operations, Number concepts, Pi, Plane geometry |
|---|---|
| Grade level | Middle School, High School, Informal Education |
| Intended audience | Learner |
| Resource type | Audio/Visual, Instructional Material, Reference Material |
| Resource format | image, image/gif, text, text/html |
| Rights | Copyright 1996-2003 WGBH Educational Foundation |
Found in collection(s)
Click on the logo to get more information about the collection.
Content contained within the resource
Support provided by Ancient Worlds Body + Brain Evolution Military + Espionage Nature Physics + Math Planet Earth Space + Flight Tech + Engineering Print Share A A A Approximating Pi By Rick Groleau Posted 09.01.03 NOVA Around 250 B.C., the Greek mathematician Archimedes calculated the ratio of a circle's circumference to its diameter. A precise determination of pi, as we know this ratio today, had long been of interest to the ancient Greeks, who strove for precise mathematical proportions in their architecture, music, and other art forms. Close approximations of pi had been known for over 1,000 years. Archimedes’ value, however, was not only more accurate, it was the first theoretical, rather than measured, calculation of pi. Here, try Archimedes' approach yourself. Launch Interactive Printable Version The Greek mathematician Archimedes used a fairly simple geometrical approach to estimate pi. See how he did it. This feature originally appeared on the site for the NOVA program Infinite Secrets. More on the method Archimedes' method finds an approximation of pi by determining the length of the perimeter of a polygon inscribed within a circle (which is less than the circumference of the circle) and the perimeter of a polygon circumscribed outside a circle (which is greater than the circumference). The value of pi lies between those two lengths. By doubling the number of sides of the hexagon to a 12-sided polygon, then a 24-sided polygon, and finally 48- and 96-sided polygons, Archimedes was able to bring the two perimeters ever closer in length to the circumference of the circle and thereby come up with his approximation. Specifically, he determined that pi was less than 3 1/7 but greater than 3 10/71. In the decimal notation we use today, this translates to 3.1429 to 3.1408. That's pretty close to the known value of 3.1416. (For simplicity's sake, we round off all figures to four decimal places.) Like Archimedes' approach, our interactive doesn't rely on specific measurements. The diameter of the circle is given an arbitrary value of 1; it doesn't matter if that number represents an inch, a foot, or a light-year. Also like Archimedes' approach, the interactive determines the length of a side of each triangle, relative to the diameter, based on the angle opposing the side being measured. Our interactive differs from Archimedes' approach in three key ways, however. First, it makes use of algebra and modern trigonometry, which were unknown in Archimedes' day—Archimedes used geometry instead. For example, he knew the ratio between one line and another in certain triangles and with this knowledge was able to figure out the length of the perimeter of a hexagon. Second, we use decimal notation, which wasn't invented until hundreds of years after Archimedes' death. To work with non-whole numbers, the ancients relied on ratios. Any calculator will tell you that the square root of 3 is 1.7321. For Archimedes, that value was 265/153 (which equals 1.7320 in decimal notation). Finally, our interactive increases the number of sides of the hexagon to 96 by increments of 1 rather than by the doublings Archimedes used. The idea is to give you a clearer sense of how ever closer in length to the perimeter of the circle the length of the hexagonal perimeters becomes with each added side. It is interesting to note that even today pi cannot be calculated precisely—there are no two whole numbers that can make a ratio equal to pi. Mathematicians find a closer approximation every year—in 2002, for example, experts at the University of Tokyo Information Technology Center determined the value of pi to over one trillion decimal places. But this is academic: the value determined by Archimedes over 2,000 years ago is sufficient for most uses today. Credits Illustrations (all) © WGBH/NOVA Related Links Contemplating Infinity If you're not mathematically inclined, the concept can mess with your mind. Working With Infinity For mathematicians, using infinity is all in a day’s work, says Stanford scholar Reviel Netz in this interview. Wisdom of the Crowds Ask enough people to estimate something, and the average of their guesses will get you close to the right answer. Andrew Wiles on Solving Fermat Wiles describes his career-long quest to prove Fermat's Last Theorem, the world's most famous mathematical problem. TV Schedule Teachers Shop NOVA Search NOVA Beta Physics & Math Resources Text (45) Describing Nature With Math How do scientists use mathematics to define reality? And why? On Being a Physicist Columbia's Brian Greene explains why history’s greatest scientific thinkers, from Newton to Feynman, became his heroes. Spooky Action at a Distance That’s what Einstein called it, dismissively, but ultra-strange quantum entanglement does exist, Brian Greene writes. Special Relativity in a Nutshell Brian Greene explains Einstein’s notion of the mutability of space and time in a way you can readily understand it. Riddles of the Multiverse USC theoretical physicist Clifford Johnson contemplates the controversial notion of multiple universes. Video (28) Light My Fire What do you get when you introduce a chunk of sodium to a Bunsen burner? Sizzling romance. The Amazing Atomic Clock What are atomic clocks? And why do we need them? Quantum Confidential Learn how to send top-secret messages using quantum mechanics. Fabric: What Is Space? The Fabric of the Cosmos, Hour 1: Surprising clues indicate that space is very much something and not nothing. Fabric: Universe or Multiverse? The Fabric of the Cosmos, Hour 4: Is our universe unique, or could it be just one in an endless “multiverse”? Multimedia (29) A Cosmic Crossword Try your hand at NOVA’s first crossword puzzle. It’s fast, physics-based, and fun. A Trip Through Spacetime Join Albert Einstein on a high-speed cab ride to see why space and time are truly in the eye of the beholder. It's Elemental In this interactive periodic table, explore the elements and their properties and abundances. Physics of Stone Arches See if you can build a cathedral arch without it collapsing, and learn more about the forces at work. Design a Fractal Create and save your own wildly colorful fractals using our generator. Audio (10) Steven Weinberg on Space The concept of “space” is a tough one to explain, even for a Nobel prize-winning physicist. Peter Galison on Time Physicist Peter Galison says that humans sense time as moving constantly forward, but it doesn’t really work that way. Sean Carroll on Time Physicist Sean Carroll says there's no such thing as past or future in the elementary laws of physics. Max Tegmark on Time Physicist Max Tegmark says that time is still one of the biggest mysteries in physics. Janna Levin on Space and Time Physicist Janna Levin says that Einstein and Newton had very different ideas about what space and time really are. NOVA Education Close You need the Flash Player plug-in to view this content. Home About this Beta About NOVA Credits Shop NOVA FAQs Newsletter PBS Privacy Policy RSS Feed Feedback Support NOVA Education Corporate Sponsorship This website was produced for PBS Online by WGBH. Website © 1996–2011 WGBH Educational Foundation Funding for NOVA is provided by David H. Koch, the Howard Hughes Medical Institute, the Corporation for Public Broadcasting, Lockheed Martin Corporation, and PBS viewers.