Update: October 24, 2022 – This article, originally written in 2016, has been updated to reflect the latest values for NASA’s Voyager 1 spacecraft, which continues to venture farther into interstellar space. The author, Marc Rayman, has ventured on too, from the chief engineer for NASA’s Dawn mission, which concluded successfully in 2018, to the chief engineer for mission operations and science at NASA’s Jet Propulsion Laboratory.

This graphic shows more than 500 of the infinite number of decimals in pi. Image credit: NASA/JPL-Caltech | + Expand image

We received this question from a fan on Facebook who wondered how many decimals of the never-ending mathematical constant pi (π) NASA-JPL scientists and engineers use when making calculations:

“Does JPL only use 3.14 for its pi calculations? Or do you use more decimals, like say [360 or even more]?”

Here’s JPL’s Chief Engineer for Mission Operations and Science, Marc Rayman, with the answer:

Thank you for your question! This isn't the first time I've heard a question like this. In fact, it was posed many years ago by a sixth-grade science and space enthusiast who was later fortunate enough to earn a doctorate in physics and become involved in space exploration. His name was Marc Rayman.

To start, let me answer your question directly. For JPL's highest accuracy calculations, which are for interplanetary navigation, we use 3.141592653589793. Let's look at this a little more closely to understand why we don't use more decimal places. I think we can even see that there are no physically realistic calculations scientists ever perform for which it is necessary to include nearly as many decimal points as you asked about. Consider these examples:

1. The most distant spacecraft from Earth is Voyager 1. As of this writing, it’s about 14.7 billion miles (23.6 billion kilometers) away. Let’s be generous and call that 15 billion miles (24 billion kilometers). Now say we have a circle with a radius of exactly that size, 30 billion miles (48 billion kilometers) in diameter, and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded to the 15th decimal, as I gave above, that comes out to a little more than 94 billion miles (more than 150 billion kilometers). We don't need to be concerned here with exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi. In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off. It turns out that our calculated circumference of the 30-billion-mile (48-billion-kilometer) diameter circle would be wrong by less than half an inch (about one centimeter). Think about that. We have a circle more than 94 billion miles (more than 150 billion kilometers) around, and our calculation of that distance would be off by no more than the width of your little finger.


Put your pi math skills to the test with this problem from NASA's Pi Day Challenge. Can you use pi to determine what fraction of a signal from Voyager 1 reaches Earth? Image credit: NASA/JPL-Caltech | + Expand image | › View lesson page

2. We can bring this closer to home by looking at our planet, Earth. It is more than 7,900 miles (12,700 kilometers) in diameter at the equator. The circumference is roughly 24,900 miles (40,100 kilometers). That's how far you would travel if you circumnavigated the globe – and didn't worry about hills, valleys, and obstacles like buildings, ocean waves, etc. How far off would your odometer be if you used the limited version of pi above? The discrepancy would be the size of a molecule. There are many different kinds of molecules, of course, so they span a wide range of sizes, but I hope this gives you an idea. Another way to view this is that your error by not using more digits of pi would be more than 30,000 times thinner than a hair!


Image credit: NASA | + Expand image

3. Let's go to the largest size there is: the known universe. The radius of the universe is about 46 billion light years. Now let me ask (and answer!) a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom, the simplest atom? It turns out that 37 decimal places (38 digits, including the number 3 to the left of the decimal point) would be quite sufficient. Think about how fantastically vast the universe is. It’s certainly far beyond what you can see with your eyes even on the darkest, most beautiful night of sparkling stars. It’s yet farther beyond the extraordinary vision of the James Webb Space Telescope. And the vastness of the universe is truly far, far, far beyond what we can even conceive. Now think about how incredibly tiny a single atom is. Isn’t it amazing that we wouldn’t need to use many digits of pi at all to cover that entire unbelievable range?

If you were to hold a single grain of sand at arm's length, you could cover the entire area of space taken up by this image, which was captured by the James Webb Space Telescope and contains thousands of galaxies. The oldest-known galaxy identified in the image is 13.1 billion years old. Image credit: NASA, ESA, CSA, STScI | + Expand image | › More about the image | Text description (PDF)

Pi is an intriguing number with interesting mathematical properties. It’s fun to think about its truly endless sequence of digits, and it may be surprising how often it appears in the equations scientists and engineers use. But there are no questions – prosaic or esoteric – in humankind’s noble efforts to explore or comprehend the marvels of the cosmos, from the unimaginably smallest scales to the inconceivably largest, that could require very many of those digits.

Hear more from Marc in his inspiring TEDx talk, “If It Isn’t Impossible, It Isn’t Worth Trying” and in his Dawn Journal, where he wrote frequent updates about the Dawn mission’s extraordinary extraterrestrial expedition to the protoplanet Vesta and dwarf planet Ceres.

Explore More

Educator Resources

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  NASA Pi Day Challenge

  This collection of illustrated math problems gets students using pi like NASA scientists and engineers exploring Earth and space.

  Grades 4-12

  Time Varies

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  Pi in the Sky Lessons

  Find everything you need to bring the NASA Pi Day Challenge into the classroom, including printable handouts of each illustrated math problem.

  Grades 4-12

  Time Varies

Articles

• 
  18 Ways NASA Uses Pi

  Whether it's sending spacecraft to other planets, driving rovers on Mars, finding out what planets are made of or how deep alien oceans are, pi takes us far at NASA. Find out how pi helps us explore space.

• 
  10 Ways to Celebrate Pi Day With NASA on March 14

  Find out what makes pi so special, how it’s used to explore space, and how you can join the celebration with resources from NASA.

Multimedia & Downloads

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  Infographic: Planet Pi

  This poster shows some of the ways NASA scientists and engineers use the mathematical constant pi (3.14) and includes common pi formulas.

Recursos en español

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  18 Maneras en Que la NASA Usa Pi

  Pi nos lleva lejos en la NASA. Estas son solo algunas de las formas en que pi nos ayuda a explorar el espacio.

TAGS: Pi, Pi Day, Dawn, Voyager, Engineering, Science, Mathematics

• 

  ABOUT THE AUTHOR

  NASA/JPL Edu

  NASA/JPL Edu supports science, technology, engineering and mathematics (STEM) education by providing NASA and JPL activities, resources and materials to educators and students in grades K-12 as well as STEM internships and research opportunities to higher education students and faculty.

Why March 14?

Pi is what’s known as an irrational number, meaning its decimal representation never ends and it never repeats. It has been calculated to more than one trillion digits, but 3.14 is often a precise enough approximation, hence the celebration occurring on March 14, or 3/14 (when written in US month/day format). The first known celebration occurred in 1988, and in 2009, the US House of Representatives passed a resolution designating March 14 as Pi Day and encouraging teachers and students to celebrate the day with activities that teach students about pi.

Why It’s Important

Pi Day is lots of fun, but its importance lies in the role that pi plays in the everyday work of scientists and engineers at JPL.

Fred Calef, a geospatial information scientist at JPL, uses pi to make measurements – like perimeter, area and volume – of features on Mars. “I use pi to measure the circularity of features, or how round or compact they are," said Calef. "Craters become more elliptical if the projectile hits the surface at a lower angle, so I use pi to measure how round a crater is to see if it impacted at a low angle.”

"We use pi every day commanding rovers on Mars," said Hallie Gengl, a rover planner for the Mars Exploration Rover Opportunity, "Everything from taking images, turning the wheels, driving around, operating the robotic arm, and even talking to Earth.”

Bryana Henderson, who specializes in planetary ices, uses lasers to explode ice samples and study their composition. "I use pi to calculate the width of my laser beam, which in turn can be used to calculate the amount of energy, or fluence, that hits my ice sample," said Henderson. "A larger fluence equals a bigger explosion in the ice, so this is a very important parameter for us."

The Pi Day Challenge

JPL has released the third installment of its popular Pi Day challenge, which gives students and the public a chance to put their pi skills to the test to solve some of the same problems NASA scientists and engineers do. The set of four illustrated math problems are compiled into a graphic (as well as classroom handouts) designed for students in grade 4 through high school – but fun for all!

This year’s problem set shows how pi can be used to map the surface of Saturn’s hazy moon Titan, track the Mars Reconnaissance Orbiter as it explores the Red Planet, keep Earth’s satellites powered as Mercury transits the sun, and put the Juno spacecraft into orbit around Jupiter.

“For Pi Day, we like to give students and the public a glimpse into how math is used at JPL through questions that feature current events involving our space missions,” said Ota Lutz, an education specialist at JPL who helped create the problem set. “For instance, to put the Juno spacecraft into orbit around Jupiter on July 4, engineers will have to slow the spacecraft just the right amount. In the Pi Day challenge, students use pi to calculate that change in velocity.”

In the challenge, students will also use pi to calculate how much sunlight is blocked by our solar system’s innermost planet as it passes between Earth and the sun. This year, Pi Day comes just a few months before the May 9 transit of Mercury, making this a timely problem.

On March 16, the answers to all four problems and the steps needed to find those answers will be released in a companion infographic on the Pi Day challenge activity page.

In addition to the Pi Day challenge, JPL is inviting the public to share their Pi Day pictures and stories online. On March 14, JPL will join in on the fun with Pi Day photos and stories from the lab.

To see a compilation of all 12 Pi Day challenge questions optimized for mobile devices and screen readers, visit: http://www.jpl.nasa.gov/edu/nasapidaychallenge

Explore More

Pi Day Challenges

• Pi in the Sky
  
• Pi in the Sky 2
  
• Pi in the Sky 3
  
• Pi in the Sky Challenge (slideshow for students)
  

Multimedia

• Planet Pi
  

Facts and Figures

• Mercury
  
• Mars
  
• Titan (moon of Saturn)
  
• Jupiter
  

Missions

• Mars Reconnaissance Orbiter
  
• Juno
  

Websites

• Mars Exploration
  
• Solar System Exploration
  
• Global Climate Change

TAGS: pi, pi day, math, mathematics

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  ABOUT THE AUTHOR

  Lyle Tavernier, Educational Technology Specialist, NASA-JPL Education Office

  Lyle Tavernier is an educational technology specialist at NASA's Jet Propulsion Laboratory. When he’s not busy working in the areas of distance learning and instructional technology, you might find him running with his dog, cooking or planning his next trip.