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Module 9
The Volume of Spheres and Cylinders

Using Spheres and Cylinders to Estimate the Volumes of Comets, Asteroids and Spacecraft

Objectives: Students will learn more about asteroids and comets through reading a NASA press release and viewing a NASA eClips video segment. Then, students will estimate and calculate volumes of comets, asteroids, and spacecraft.

Mathematics Skill or Topic Area:

The Volume of Spheres and Cylinders

Next Gen Science Standards: ESS1: Earth’s Place in the Universe

Common Core ELA for Science: RST.6-8.2. Determine the central ideas or conclusions of a text; provide an accurate summary of the text distinct from prior knowledge or opinions. RST.6-8.8. Distinguish among facts, reasoned judgment based on research findings, and speculation in a text. RST.6-8.9. Compare and contrast the information gained from experiments, simulations, video, or multimedia sources with that gained from reading a text on the same topic.

Common Core Math Standard: CC.7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.

Video Engagement: Comets - It's Done With Math
Find out how NASA engineers use a repurposed spacecraft to return to a comet for a second visit to uncover secrets about the formation of the solar system. The math concept of angular size is introduced in this video. If you choose, have your students review how to use angular size as a tool to calculate the size of a comet. (5 minutes) View Program

Engage your students with a press release:

NASA's Dawn Spacecraft Begins Exploration of Vesta

NASA's Dawn spacecraft, the first ever to orbit an object in the main asteroid belt, is spiraling towards its first of four intensive science orbits. That initial orbit of the rocky world Vesta begins Aug. 11, at an altitude of nearly 1,700 miles (2,700 kilometers) and will provide in-depth analysis of the asteroid. Vesta is the brightest object in the asteroid belt as seen from Earth and is thought to be the source of a large number of meteorites that fall to Earth.

This image was taken at a distance of 3,200 miles (5,200 kilometers). Images from Dawn's framing camera, taken for navigation purposes and as preparation for scientific observations, are revealing the first surface details of the giant asteroid. These images go all the way around Vesta, since the giant asteroid turns on its axis once every five hours and 20 minutes.

"Now that we are in orbit around one of the last unexplored worlds in the inner solar system, we can see that it's a unique and fascinating place," said Marc Rayman, Dawn's chief engineer and mission manager at NASA's Jet Propulsion Laboratory in Pasadena, Calif.

After traveling nearly four years and 1.7 billion miles (2.8 billion kilometers), Dawn has been captured by Vesta's gravity, and there currently are 1,800 miles (2,900 kilometers) between the asteroid and the spacecraft. The giant asteroid and its new neighbor are approximately 114 million miles (184 million kilometers) away from Earth.

"We have been calling Vesta the smallest terrestrial planet," said Chris Russell, Dawn's principal investigator at UCLA. "The latest imagery provides much justification for our expectations. They show that a variety of processes were once at work on the surface of Vesta and provide extensive evidence for Vesta's planetary aspirations."

Press release date line - August 1, 2011

Press release location: [ Click Here ]

Explore math connections with


Problem I - Comparing the Volume of Vesta and the Moon - Astronomers often need to know the volume of the objects they study. Planets and large asteroids are distinctly spherical in shape. Students use this fact to estimate the volume of the asteroid Vesta in comparison to our own moon. Topics include scale and the volume of a sphere. [Open PDF]

Problem II - The Magnetospheric, Multi-Scale Satellite Constellation. - Students determine how the MMS constellation satellites are stacked inside the third stage of an Atlas V rocket by using the properties of cylindrical volumes. Topics include scale and the volume of a cylinder. [Open PDF]

Explain your thinking:

Write your own problem - Using information found in the Math Connection problems, the press release or the video program, create your own math problem that involves estimating the volume of an astronomical object or spacecraft. Explain why you set the problem up this way, and how you might find its answer.

Evaluate your understanding:

Challenge Problem - In 2010, NASA's Deep Impact spacecraft flew by the nucleus of Comet Hartley-2. The images showed that its shape was dumbell-like, and could be approximated as a cylinder with a diameter of 0.5 kilometers and a height of 0.25 kilometers, and two spheres with diameters of 0.8 kilometers. About what is the volume of Hartley-2 in cubic kilometers? Explain how you arrived at this answer.

Answer: Using the formula for a sphere with a radius of 0.4 kilometers, the volume of each sphere is V = 0.27 cubic kilometers. The volume of the cylinder is just V = 0.05 cubic kilometers, so the total volume is V = 2 (0.27)+0.05 = 0.59 cubic kilometers.



3-D Solar System

Extend your new knowledge - Students use the Eyes on the Solar System simulator to study the volumes of spacecraft including the Hubble Space Telescope, CALIPSO and Kepler. They design a new Hubble Space Telescope with a larger volume and determine its mirror diameter. [ Open PDF ]