Celestial Mechanics
Geometry in Space
Lesson Plan
Subject:
Geometry
Grade Level:
10
Author:
Stephanie Boyd
Project Partners: E3 Teacher Summer Research Program
Department of Aerospace Engineering
Texas A&M University
Celestial Mechanics Lesson Plans - 2
Purpose
The purpose of this unit is to provide high school geometry students with an answer to
the question, “Why do I need to learn this?” Each set of lesson plans materials connects
TEKS and NCTM standards with applications in current aerospace engineering research.
The goal is to interest the student by providing relevant real-world examples to geometry
principles.
Topics Covered
Exploring Circles
Angles and Arcs
Equations of Circles
Objectives
Each lesson plan contains objectives. The additional instructional materials are designed
to help the teacher and students reach the objectives.
Texas Essential Knowledge and Skills (TEKS)
Each lesson plan addresses applicable standards from the TEKS. The TEKS are
governed by the Texas Education Agency (TEA) and represent the objectives that will
appear on the end of course test, the Texas Assessment of Knowledge and Skills (TAKS).
NCTM Standards
Each lesson plan covers standards from the National Council of Teachers of Mathematics
(NCTM).
Background
Every day, professors at Texas A&M University use geometry in their research! In the
field of aerospace engineering, there is a topic called “Celestial Mechanics”. What does
this mean? Celestial translates to “space” and mechanics to “workings or machinery”.
So, this topic deals with how things work in space! One part of celestial mechanics
involves machines in space, like satellites! These researchers find ways to improve the
quality, efficiency, and cost of satellites.
Did you know that one satellite launch could cost up to $400 million? What if NASA
only has $50 million to use on each satellite? Then these researchers test lighter
materials and different structures to help decrease the launch cost. A major research
problem in all fields of engineering is to minimize the costs of putting all of the new
ideas and technologies into action. In other words, the goal is to make the costs as low as
possible. So, where does geometry fit into this? You’ll find out in the lessons!
Celestial Mechanics Lesson Plans - 3
Lesson One – Exploring Circles
Objectives:
The student will identify and use parts of circles.
The student will solve problems involving the circumference of a circle.
Resources:
Glencoe: Geometry, 1998 ed. (Chapter 9, Section 1)
Materials:
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

Calculator
Application – transparency
Guided Practice – transparency, individual copies for each student
TEKS:
1.a., 1.b., 2.b., 4.a., 7.a., 9.b., 9.c.
NCTM Standards:
1, 2, 3, 4, 7
Applicable Problem:
Satellites are used for many different reasons! One type of satellite is used to take
pictures of Earth from space. If we look at a picture of the Earth taken from a satellite,
what shape is it? It looks like a circle! We know that the Earth is actually a 3dimensional object, but from a 2-dimensional perspective, it looks like a circle! We can
use the properties of a circle – radius, diameter, and circumference – to describe Earth.
The Earth even has a center point! We need to know this information to launch a satellite
into space so that it will orbit, or travel around, the Earth.
Demonstration and Guided Practice:


How do we get to see pictures of the Earth?
o Display Exploring Circles: Application
What is a circle? What are the parts of a circle?
o Definitions: circle, radius, chord, diameter, circumference
 Circle: the set of all points in a plane that are a given distance
from the center of the circle
 Radius: a segment that has one endpoint at the center of the circle
and the other endpoint on the circle
 Chord: a segment that has its endpoints on the circle
 Diameter: a chord that contains the center of the circle
 Circumference: the distance around the circle
o Can we point out the parts of a circle on the satellite picture of Earth?
Celestial Mechanics Lesson Plans - 4
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
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Ask students to draw and label the different parts of a circle on the
overhead picture of Earth.
How are the parts of a circle related?
o Equations: C = circumference, r = radius, d = diameter
 C = 2πr = πd
o How are the radius and diameter of a circle related?
o What is the approximation of π?
Display and complete Exploring Circles: Guided Practice
Independent Practice:
Ask each student to find examples of circular at home or in the classroom and measure
different parts of the circles. Do the formulas and relationships hold? Assign additional
problems from the text or other sources as desired.
Celestial Mechanics Lesson Plans - 5
Exploring Circles: Application
Satellite Pictures of the Earth
Digital picture taken from a geostationary satellite: GOES (Geostationary Operational
Environmental Satellite)
http://antwrp.gsfc.nasa.gov/apod/ap000420.html
Other Links
http://visibleearth.nasa.gov/
Searchable directory of images, visualizations, and animations of the Earth taken from
satellites
http://www.nasa.gov
Helpful information for educators and students
http://www.goes.noaa.gov/
Images of the Earth taken from geostationary satellites
http://www.howstuffworks.com/satellite.htm
Great resource for learning about satellites and tons of other concepts
Celestial Mechanics Lesson Plans - 6
Exploring Circles: Guided Practice
1. The radius of Earth is 6378 kilometers. What is the diameter and circumference
of Earth (in kilometers)?
2. The properties of a circle can also be used to describe the moon, Earth’s natural
satellite! The circumference of the moon is approximately 10927 km. What is
the radius and diameter of the moon?
Celestial Mechanics Lesson Plans - 7
3. Now, we can make the same calculations for any circle! The chord AB is 5 cm
and chord BC is 12 cm, and ABC is a right triangle. What is the length of AC, the
diameter of circle D? What is the length of AD? What is the circumference of
circle D?
C
D
A
B
Celestial Mechanics Lesson Plans - 8
Exploring Circles: Guided Practice - Solutions
1. The radius of Earth is 6378 kilometers. What is the diameter and circumference
of Earth (in kilometers)?
D = 2r = 2(6378) = 12756 km
C = 2πr = 12756π ≈ 40074 km
2. The properties of a circle can also be used to describe the moon, Earth’s natural
satellite! The circumference of the moon is approximately 10927 km. What is
the radius and diameter of the moon?
C = 10927 = 2πr
r = C / 2π ≈ 1739 km
3. Now, we can make the same calculations for any circle! The chord AB is 5 cm
and chord BC is 12 cm, and ABC is a right triangle. What is the length of AC, the
diameter of circle D? What is the length of AD? What is the circumference of
circle D?
C
D
A
B
Using the Pythagorean theorem with AC as the hypotenuse of a right triangle, AC = 13.
Therefore, the length of AD = 13/2 = 6.5. The circumference of D = πd = π(13) ≈ 40.84
Celestial Mechanics Lesson Plans - 9
Lesson Two: Angles and Arcs
Objectives:
The student will recognize major arcs, minor arcs, semicircles, and central angles.
The student will find measures of arcs and central angles.
Resources:
Glencoe: Geometry, 1998 ed. (Chapter 9, Section 2)
Materials:
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

Calculator
Application – transparency
Guided Practice – transparency, individual copies for each student
TEKS:
1.a., 2.b., 4.a., 5.a., 8.b., 9.b., 9.c
NCTM Standards:
1, 2, 3, 4, 7, 10
Applicable Problem:
Do you have a cell phone? Have you ever watched satellite TV on the DISH or
DIRECTV network? These things wouldn’t exist without the technology that lets us put
satellites into space! When satellites are launched into space, they are put into an orbit
around the Earth. One special kind of orbit is called the geostationary orbit, GSO for
short. This is the orbit that communications satellites use, because the orbit is circular.
Because of this, a GSO satellite will stay exactly the same distance away from the Earth
all the time. In fact, the GSO satellite even stays above the same point on the Earth all
the time because it rotates at the same rate that the Earth does. We can use what we
know about circles to describe these satellite orbits!
Demonstration and Guided Practice:
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
What exactly is the GSO (geostationary orbit)?
o Display Angles and Arcs: Application
Definitions: central angle, arc, minor arc, major arc, semicircle, adjacent arcs, arc
length, concentric circles, congruent circles, congruent arcs
o Central angle: an angle whose vertex is at the center of a circle
o Arc: an unbroken part of a circle
o Minor arc: consists of all points interior to two points on a circle; its
measure is always less than 180 degrees; it is named by the two points on
the circle that define it
Celestial Mechanics Lesson Plans - 10
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
o Major arc: consists of all points exterior to two points on a circle; its
measure is always greater than 180 degrees; it is named by three points on
the circle that define it
o Semicircle: an arc whose measure is 180 degrees
o Adjacent arcs: arcs of a circle that have exactly one point in common; the
measures of adjacent arcs can be added to find the measure of the arc
formed by the adjacent arcs
o Arc length: the part of the circumference of a circle that is proportional to
the measure of the central angle when compared to the entire circle
o Concentric circles: circles that lie in the same plane and have the same
center but different radii
o Congruent circles: circles that have the same radius
o Congruent arcs: two arcs of the same circle that have the same measure;
these arcs also have the same arc length
Important Concepts:
o Sum of central angles: The sum of the measures of the central angles of a
circle with no interior points in comment is 360.
o The measure of a minor arc is the measure of its central angle.
o The measure of a major arc is 360 minus the measure of its central angle.
o The measure of a semicircle is 180.
o The measure of an arc formed by two adjacent arcs is the sum of the
measures of the two arcs.
Display Angles and Arcs: Guided Practice
Independent Practice:
Ask the students to create their own arc measure/arc length problem to solve. Have them
design a problem and then trade with another student so that the students get experience
working multiple varieties of problems.
Celestial Mechanics Lesson Plans - 11
Angles and Arcs: Application
1. What is a geostationary orbit?
A geostationary orbit (GSO) is a circular orbit where a satellite can rotate around a fixed
point on Earth.
2. What types of satellite are put into a geostationary orbit?
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Communication – cell phones, satellite TV, newspapers
Navigation – GPS, military, emergency situations (to locate airplanes or boats that
are in trouble)
Data-gathering – weather, observation
3. How much does it cost to launch a satellite?
It can cost anywhere from $50 - $400 million to launch each satellite!
4. How many satellites are in a geostationary orbit?
Each tick mark represents one of the hundreds of satellites in geostationary orbit!
http://www.windows.ucar.edu/spaceweather/geo_sats.html
5. How far is a GSO satellite from Earth?
The distance from a GSO satellite to Earth’s surface is approximately 35000 km. The
distance from GSO satellite to the center of the Earth is approximately 42000 km.
Celestial Mechanics Lesson Plans - 12
Angles and Arcs: Guided Practice
1. Name the following angles and arcs.
Central angle:
B
A
GSO
Minor arc:
Earth
C
Major arc:
Semicircle:
D
2. The radius of orbit around S is 9, and minor angle GO
measures 120 degrees. Find the arc length of minor arc
GO and major arc GEO.
G
S
E
120
O
Celestial Mechanics Lesson Plans - 13
3. Satellite S is in a geostationary orbit around Earth. The major arc between point A
and point B has measure 240. If S begins its orbit at point A and then travels
clockwise around the orbit to point B, how far has it traveled? (Hint: The distance
from S to the Earth at any point on the orbit is 42000 km… what can you use this
information to calculate?)
A
S
Earth
B
240
4. Since it takes a satellite in geostationary orbit about 24 hours to make a complete
rotation, how much time did it take satellite S from problem 3 to travel from A to B?
5. Based on the previous example problems, how fast must a satellite travel to stay in
GSO?
Celestial Mechanics Lesson Plans - 14
Angles and Arcs: Guided Practice - Solutions
1. Name the following angles and arcs.
Central angle: BEA, BEC, CED, DEA
B
A
GSO
Minor arc: BC, CD, DA, AB, AC
E
C
Major arc: BDC, CAD, DCA, ADB,
ADC
Semicircle: BD
D
G
2. The radius of an orbit around S is 9, and minor angle GO
measures 120 degrees. Find the arc length of minor arc
GO and major arc GEO.
S
C = 2πr = 18π
Length of GO = (120/360) * 18π = 6π ≈ 18.85
Length of GEO = (360 – 120)/360 * 18π = 12π ≈ 37.70
E
120
O
Celestial Mechanics Lesson Plans - 15
3. Satellite S is in a geostationary orbit around Earth. The major arc between point A
and point B has measure 240. If S begins its orbit at point A and then travels
clockwise around the orbit to point B, how far has it traveled? (Hint: The distance
from S to the Earth at any point on the orbit is 42000 km… what can you use this
information to calculate?)
A
S
Earth
B
240
Distance from S to center of Earth is 42000 km (GSO)
C = 2πr = 2(42000)π ≈ 264000 km
(240/360) * C ≈ 176000 km
4. Since it takes a satellite in geostationary orbit about 24 hours to make a complete
rotation, how long did it take satellite S (from problem 3) to travel from A to B?
t = travel time from A to B
t hrs / 24 hrs ≈ 176000 km / 264000 km
t ≈ 16 hrs
5. Based on the previous example problems, how fast must a satellite travel to stay in
GSO?
v = velocity of a satellite in GSO
v = 264000 km / 24 hrs ≈ 11000 km/hr
Celestial Mechanics Lesson Plans - 16
Lesson Three: Equations of Circles
Objective:
The student will write and use the equation of a circle in the coordinate plane.
Resources:
Glencoe: Geometry, 1998 ed. (Chapter 9, Section 8)
Materials:





Calculator
Application – transparency
Guided Practice – transparency, individual copies for each student
Independent Practice – transparency, individual copies for each student
Graph paper
TEKS:
1.a., 2.b., 4.a., 7.a.
NCTM Standards:
1, 2, 3, 4, 5, 7, 8
Applicable Problem:
Sometimes aerospace engineers would like to calculate the exact position of a satellite.
Have you ever seen the OnStar system in a really expensive car? The OnStar system is
used to tell you where you are located on the Earth and help you get to where you are
going without getting lost. You never have to ask for directions! The OnStar system is a
type of GPS – a Global Positioning System. How does this system work? It relies on
satellites! Satellites located in orbit around the Earth tell a GPS receiver, like the one in a
car, where it is located. To find out this information, satellites “talk” to each other.
These satellites have to know where the other satellites are located to make the GPS
system work. We can show this on the coordinate plane!
Demonstration and Guided Practice:



Can we figure out the equation of an orbit?
o Display Equations of Circles – Application
The standard equation for a circle with center at (h, k) and a radius of r units is:
o (x – h)2 + (y – k)2 = r2
Display Equations of Circles – Guided Practice
Independent Practice:

Equations of Circles – Independent Practice/Unit Project
o This project is designed for use after each of these lesson plans have been
completed. They will need time to work on the project outside of class.
Celestial Mechanics Lesson Plans - 17
Equations of Circles: Application
Sputnik: The First Satellite (1957)
A NAVSTAR GPS Satellite
(Currently in a geostationary orbit)
Celestial Mechanics Lesson Plans - 18
Equations of Circles: Guided Practice
1. Using your knowledge about the radii of Earth (approximately 6400
km) and of the geostationary orbit (approximately 42000 km from the
center of the Earth), plot Earth and the GSO on the coordinate plane.
Using the formula for the equations of circles, write the equations
that represent the surface of Earth and the GSO.
Celestial Mechanics Lesson Plans - 19
Equations of Circles: Guided Practice
2. Planet A can be represented on the coordinate plane by this
equation: (x + 2)2 + (y – 3)2 = 4. Planet B corresponds to this
equation: (x – 5)2 + (y + 5)2 = 16. Before you draw the graph of these
planets, decide which quadrant each one belongs in. Why? Then,
plot planets A and B on the coordinate plane.
Celestial Mechanics Lesson Plans - 20
Equations of Circles: Guided Practice – Solutions
1. Earth: x2 + y2 = 64002
GSO: x2 + y2 = 420002
2. A: quadrant II
B: quadrant IV
Celestial Mechanics Lesson Plans - 21
Equations of Circles: Independent Practice
The discovery:
NASA has just discovered a new planet in the universe, and it has been named Planet
Aggie! They know a few facts about it, but your task is to help NASA plot a map of this
new planet and its satellites! Here is the data:
1) If the Sun is located at the origin (0,0), the new planet is located 5 units to the east
and 12 units to the north.
2) There are 2 geostationary orbits around Planet Aggie! Each of these orbits
contains 2 satellites.
3) Orbit “Whoop” has an altitude of 3 units. Satellite “Gig” is located at (5, 15) and
Satellite “Em” is located at (8, 12).
4) Orbit “Howdy” is given by the equation (x – 5)2 + (y – 12)2 = 25. Satellite “A” is
located 12 units north of the Sun and Satellite “M” is located due south of “TX”.
Your task:
Using the “Planet Aggie” coordinate plane, plot and label the Sun, Planet Aggie, the two
orbits, and the four satellites. Then answer the following questions and show your work
on a separate sheet of paper:
1. How far is Planet Aggie from the Sun?
2. Assume that Planet Aggie revolves around the Sun in a circular orbit. What is the
equation of this orbit?
3. How far does Planet Aggie travel during each revolution on its orbit around the
Sun?
4. What is the equation associated with “Whoop”?
5. What is the distance from “Gig” to “Em” (clockwise orbit)?
6. What is the altitude (radius) of “Howdy”?
7. What are the coordinates of “A”?
8. What are the coordinates of “M”?
9. What is the distance from “A” to “M” (clockwise orbit)?
 Bonus : Reveille is an aerospace engineering student on Planet Aggie. The
number of hours in Reveille’s degree is 128, and that is equal to one revolution of
Planet Aggie. Reveille is very smart and wants to take 16 hours per semester. How
many semesters can she complete during one revolution on Planet Aggie?
Celestial Mechanics Lesson Plans - 22
Equations of Circles: Independent Practice
Celestial Mechanics Lesson Plans - 23
Equations of Circles: Independent Practice – Solutions
1. How far is Planet Aggie from the Sun? 13 units (use Pythagorean Theorem)
2. Assume that Planet Aggie revolves around the Sun in a circular orbit. What is the
equation of this orbit? x2 + y2 = 132
3. How far does Planet Aggie travel during each revolution on its orbit around the
Sun? c = 2πr = 2(13)π = 81.68 units
4. What is the equation associated with “Whoop”? (x – 5)2 + (y – 12)2 = 32
5. What is the distance from “Gig” to “Em”? (90/360) * 2(3)π = 4.71 units
6. What is the altitude (radius) of “Howdy”? 5 units
Celestial Mechanics Lesson Plans - 24
7. What are the coordinates of “A”? (0, 12)
8. What are the coordinates of “M”? (5, 8)
9. What is the distance from “A” to “M”? (360 – 90) / 360 * 2(5)π = 23.56 units
 Bonus : Reveille is an aerospace engineering student on Planet
Aggie. The number of hours in Reveille’s degree is 128, and that is
equal to one revolution of Planet Aggie. Reveille is very smart and
wants to take 16 hours per semester. How many semesters can she
complete during one revolution on Planet Aggie?
(128 hrs) / 16 (hrs/semester) = 8 semesters