Problem Set
Pi in the Sky 10
Overview
In the 10th installment of the Pi in the Sky illustrated problem set, students use pi to calculate the size of a rock sample collected by the Perseverance Mars rover, compare the sensitivity of two space telescopes, determine the composition of an asteroid, and find out how much of the Sun will be covered during a solar eclipse.
Materials
Background
![](https://mars.nasa.gov/system/resources/detail_files/26895_PIA25326-web.jpg)
This illustration shows a concept for multiple robots that would team up to ferry to Earth samples of rocks and soil being collected from the Martian surface by NASA's Mars Perseverance rover. Image credit: NASA/JPL-Caltech | › Full image and caption
![](https://svs.gsfc.nasa.gov/vis/a010000/a010700/a010776/JWST-HST_Mirror_Compare_ipod_lg.00352_print.jpg)
Image from animation comparing the relative sizes of James Webb's primary mirror to Hubble's primary mirror. Credit: NASA/Goddard Space Flight Center . | › Full animation
![](https://www.nasa.gov/wp-content/uploads/2023/03/pia24472-psyche-asteroid-illustration-16_0.jpg)
This illustration depicts the metal-rich asteroid Psyche, which is located in the main asteroid belt between Mars and Jupiter. Credits: NASA/JPL-Caltech/ASU | + Full image and caption
![](https://upload.wikimedia.org/wikipedia/commons/2/22/Solar_Eclipse_May_20%2C2012.jpg)
This image sequence shows an annular solar eclipse from May 2012. The bottom right frame illustrates the distinctive ring, or "annulus," of such eclipses. A similar eclipse will be visible from the South Pacific on May 10, 2013. Credits: Brocken Inaglory, CC BY-SA 3.0, via Wikimedia Commons | + Expand image
Tubular Tally
NASA’s Mars rover, Perseverance, was designed to collect rock samples that will eventually be brought to Earth by a future mission. Sending objects from Mars to Earth is very difficult and something we've never done before. To keep the rock cores pristine on the journey to Earth, the rover hermetically seals them inside a specially designed sample tube. Once the samples are brought to Earth, scientists will be able to study them more closely with equipment that is too large to make the trip to Mars. In Tubular Tally, students use pi to determine the volume of a rock sample collected in a single tube.
Rad Reflection
When NASA launched the Hubble Space Telescope in 1990, scientists hoped that the telescope, with its large mirror and sensitivity to ultraviolet, visible, and near-infrared light, would unlock secrets of the universe from an orbit high above the atmosphere. Indeed, their hope became reality. Hubble’s discoveries, which are made possible in part by its mirror, rewrote astronomy textbooks. In 2022, the next great observatory, the James Webb Space Telescope, began exploring the infrared universe with an even larger mirror from a location beyond the orbit of the Moon. In Rad Reflection, students use pi to gain a new understanding of our ability to peer deep into the cosmos by comparing the area of Hubble’s primary mirror with the one on Webb.
Metal Math
Orbiting the Sun between Mars and Jupiter, the asteroid (16) Psyche is of particular interest to scientists because its surface may be metallic. Earth and other terrestrial planets have metal cores, but they are buried deep inside the planets, so they are difficult to study. By sending a spacecraft to study Psyche up close, scientists hope to learn more about terrestrial planet cores and our solar system’s history. That's where NASA's Psyche comes in. The mission will use specialized tools to study Psyche's composition from orbit. Determining how much metal exists on the asteroid is one of the key objectives of the mission. In Metal Math, students will do their own investigation of the asteroid's makeup, using pi to calculate the approximate density of Psyche and compare that to the density of known terrestrial materials.
Eclipsing Enigma
On Oct. 14, 2023, a solar eclipse will be visible across North and South America, as the Moon passes between Earth and the Sun, blocking the Sun's light from our perspective. Because Earth’s orbit around the Sun and the Moon’s orbit around Earth are not perfect circles, the distances between them change throughout their orbits. Depending on those distances, the Sun's disk area might be fully or only partially blocked during a solar eclipse. In Eclipsing Enigma, students get a sneak peek at what to expect in October by using pi to determine how much of the Sun’s disk will be eclipsed by the Moon and whether to expect a total or annular eclipse.
Procedures
Tubular Tally
The Perseverance Mars rover is designed to collect rock samples that will eventually be brought to Earth for further study. This would be the first time we've ever brought back samples from Mars! After scientists identify an interesting rock they would like the rover to collect, Perseverance uses a special coring bit to drill out a rock cylinder 13 mm in diameter. As the rover drills, the rock core moves into one of 38 available tubes that will store the rock sample – sealed until it is opened one day in a lab on Earth.
If the coring bit collects a rock cylinder 60 mm in length, what is the volume of the rock in the sample tube?
› Learn more about the Mars Perseverance rover
![The center of the image shows an illustration of the Perseverance Mars rover with its robotic arm stretched out to the right, touching a rock on the ground. An inset in the lower right shows a cutaway of a coring bit filled with a rock sample measuring 13 mm wide and 60 mm tall. In the lower left a sealed sample tube is shown on the ground next to the rover. A thought bubble showing hands holding a sample tube is coming from the top of the rover's mast. The text of the Tubular Tally problem is shown on the left.](https://www.jpl.nasa.gov/edu/images/activities/piday2023_mars.png)
Image credit: NASA/JPL-Caltech | + Expand image
Rad Reflection
The James Webb Space Telescope was designed to look back at some of the earliest galaxies in the universe. To capture light from these distant and faint objects, the telescope must be very sensitive. Webb uses 18 hexagonal mirrors that combine to form a massive primary mirror with a surface area of 26.4 m2. This large mirror allows the telescope to collect incredibly faint infrared light and reflect it onto four onboard science instruments, like the Mid-Infrared Instrument, or MIRI. This science instrument can reveal stars hidden within gas and dust clouds and tell scientists about the materials that make up distant galaxies.
Launched in 1990, the Hubble Space Telescope changed our understanding of the universe when it began operations using a primary mirror that had a diameter of just 2.4 meters.
How much bigger is the area of Webb’s primary mirror than Hubble’s?
› Learn more about the Webb Telescope
![In the upper left of the image is an illustration of the James Webb Space Telescope. In the lower left is an illustration of the Hubble Space Telescope. To the right of each telescope is an arrow pointing to a face-on view of its primary mirror. Next to the Webb telescope is a primary mirror composed of 18 hexagonal gold-plated mirrors arranged in a roughly circular shape. Above the mirror is text stating the surface area is 26.4 square meters. Next to the Hubble telescope is a round primary mirror with text stating the diameter is 2.4 meters. Both telescopes are in front of an illustrated star field. On the right hand side of the image is a view of space containing stars, spiral galaxies, and elliptical galaxies. Some of the galaxies are warped as a result of gravitational lensing. The text of the Rad Reflection problem is shown on the right.](https://www.jpl.nasa.gov/edu/images/activities/piday2023_jwst.png)
Image credit: NASA/JPL-Caltech | + Expand image
NASA's Universe of Learning materials are based upon work supported by NASA under award number NNX16AC65A to the Space Telescope Science Institute, working in partnership with Caltech/IPAC, Center for Astrophysics | Harvard & Smithsonian, and the Jet Propulsion Laboratory.
Metal Math
Asteroid (16) Psyche is of particular interest to scientists because ground-based observations indicate that the surface may be metallic. Earth and other terrestrial planets have metal cores, but they are buried deep inside the planets, so they are difficult to study. If Psyche consists of a large amount of metal, it might resemble a planetary core from which we could learn about terrestrial planet core formation. Determining how much metal exists on the asteroid is one of the goals of NASA’s Psyche mission, which will use specialized tools to study the asteroid's composition from orbit.
Psyche has a roughly triaxial ellipsoid shape with axes of about 290 km, 245 km, and 170 km. Its mass, as estimated from its gravitational effects on nearby bodies such as Mars, is about 2.7 x 1019 kg. Use the formula for volume, V = 4/3 πabc, where a, b, and c are the lengths of the semi-axes, to compute Psyche's approximate density.
Based on the average density of terrestrial materials (listed below), does Psyche's density support the observations indicating the presence of metal?
Average density of terrestrial materials
- ice: 917 kg/m3
- water: 997 kg/m3
- rock: 1,600 - 3,500 kg/m3
- metal: 534 - 22,590 kg/m3
› Learn more about the Psyche mission
![In the lower right of the image is an illustration of a telescope inside an observation dome. Next to the telescope is a computer screen displaying text that reads 'analyzing asteroid makeup...comparing density to...ice: 917 kg/m^3, water: 997 kg/m^3, rock: 1600-3500 kg/m^3, metal: 534-22,590 kg/m^3. Result.' In the center of the image is an inset of an illustrated asteroid (16) Psyche. The asteroid has the upper right quarter cut out to show internal semi-major axis lines. On the right of the image is the Psyche spacecraft in front of a field of stars. The text of the Metal Math problem is shown on the left.](https://www.jpl.nasa.gov/edu/images/activities/piday2023_psyche.png)
Image credit: NASA/JPL-Caltech | + Expand image
Eclipsing Enigma
A solar eclipse occurs when the Moon passes between Earth and the Sun, fully or partially blocking the Sun's light from our perspective. Because Earth’s orbit around the Sun and the Moon’s orbit around Earth are not perfect circles, the distances between them change throughout their orbits. During a total eclipse, the distances are such that the Moon covers all of the Sun's disk area. When the Moon is farther from Earth during an eclipse, it leaves a glowing ring of sunlight shining around the Moon, resulting in an annular eclipse.
On Oct. 14, 2023, a solar eclipse will be visible across North and South America. The Sun, with a radius of 695,700 km, will be 148,523,036 km from Earth. The Moon, with a radius of 1,737 km, will be 388,901 km from Earth.
What percentage of the Sun’s disk area will be obscured by the Moon? Will the eclipse be an annular eclipse or total eclipse?
› Learn more about the 2023 eclipse
![On the left side of the illustration is the sun. Two yellow light beams from the top of the sun and two from the bottom of the sun extend diagonally toward the lower right and upper right, respectively, where they intercept the moon before continuing to Earth. The moon is shown illuminated on the half facing the sun and shadowed on the side opposite the sun. The region within the light beams becomes darker to indicate shadow between the moon and Earth. A wide dark swath across Earth indicates where a partial solar eclipse will be visible while a narrow, darker swath shows where the full eclipse will be visible. An inset in the lower left shows a person wearing eclipse glasses looking toward the sun. A reflection of the eclipse is shown in the glasses. Text above the inset reads 'Remember: never look directly at the Sun without proper eye protection.' The text of the Eclipsing Enigma problem is shown on the right.](https://www.jpl.nasa.gov/edu/images/activities/piday2023_eclipse.png)
Image credit: NASA/JPL-Caltech | + Expand image
![Infographic of all of the Pi in the Sky 10 graphics and problems](https://www.jpl.nasa.gov/edu/images/activities/piday2023_infographic.png)
Image credit: NASA/JPL-Caltech | + Expand image
Assessment
![Illustrated answer key for the Pi in the Sky 10 Math Problem Set](https://www.jpl.nasa.gov/edu/images/activities/piday2023_answers.png)
Image credit: NASA/JPL-Caltech | + Expand image
Extensions
Participate
Join the conversation and share your Pi Day Challenge answers with @NASAJPL_Edu on social media using the hashtag #NASAPiDayChallenge
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Pi Day Challenge Lessons
Here's everything you need to bring the NASA Pi Day Challenge into the classroom.
Grades 4-12
Time Varies
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NASA Pi Day Challenge
In this challenge, students can use pi to solve some of the same problems faced by NASA scientists and engineers.
Grades 4-12
Time Varies
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Pi Day: What's Going 'Round
Tell us what you're up to this Pi Day and share your stories and photos with NASA.
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National Council of Teachers of Mathematics: Notice and Wonder
Creative brainstorming through noticing and wondering encourages student participation, engagement, and students' understanding of the NASA Pi Day Challenge.
Subject Mathematics
Blogs and Features
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How Many Decimals of Pi Do We Really Need?
While you may have memorized more than 70,000 digits of pi, world record holders, a JPL engineer explains why you really only need a tiny fraction of that for most calculations.
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Slideshow: 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.
Related Lessons for Educators
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Robotic Arm Challenge
In this challenge, students will create a model robotic arm to move items from one location to another. They will engage in the engineering design process to design, build and operate the arm.
Grades K-8
Time 30 min to 1 hour
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NASA's Mission to Mars Student Challenge
Take part in the exploration of Mars and bring students along for the ride with NASA's Perseverance rover.
Grades K-12
Time Varies
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Moon Phases
Students learn about the phases of the moon by acting them out.
Grades 1-6
Time 30 min to 1 hour
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Modeling the Earth-Moon System
Students learn about scale models and distance by creating a classroom-size Earth-Moon system.
Grades 6-8
Time 30 min to 1 hour
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Math of the Expanding Universe
Students will learn about the expanding universe and the redshift of lightwaves, then perform their own calculations with a distant supernova.
Grades 9-12
Time 30 min to 1 hour
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The Expanded Universe: Playing with Time Activity Guide
In this activity, participants use balloons to model the expansion of the universe and observe how expansion affects wavelengths of light and distance between galaxies
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James Webb Space Telescope STEM Toolkit
Find a collection of resources, activities, videos, and more for your students to learn about NASA’s newest space observatory.
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Modeling an Asteroid
Lead a discussion about asteroids and their physical properties, then have students mold their own asteroids out of clay.
Grades 3-5
Time 30 min to 1 hour
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Math Rocks: A Lesson in Asteroid Dynamics
Students use math to investigate a real-life asteroid impact.
Grades 8-12
Time 30 min to 1 hour
Related Activities for Students
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How to Make a Pinhole Camera
Learn how to make your very own pinhole camera to safely see a solar eclipse in action!
Type Project
Subject Engineering
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Collection: Exploring Mars
Make a cardboard rover, design a Mars exploration video game and explore more STEM projects, slideshows and videos for students.
Type Project
Subject Science
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What's That Space Rock?
Find out how to tell the difference between asteroids, comets, meteors, meteorites and other bodies in our solar system.
Type Slideshow
Subject Science
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10 Things We Can Learn from Webb's First Images
Take a closer look at how images from NASA's most powerful space telescope yet are helping to answer some of astronomers' most burning questions.
Type Slideshow
Subject Science
Multimedia
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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.
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Downloads
Can't get enough pi? Download this year's NASA Pi Day Challenge graphics, including mobile phone and desktop backgrounds:
Recursos en español
Facts and Figures
Websites
- Webb Space Telescope
- Mars Exploration
- Perseverance Mars Rover
- Mars Sample Return
- Psyche Mission
- MIRI Instrument
- 2023 Eclipse