www.nasa.gov

National Aeronautics and 

Space Administration

NASA Space Communications 

and Navigation

CCSS.MATH.CONTENT.HSG.C.A.3

ACTIVITY SHEET

Geometry of Circles GRADES 9–10

PAGE 1

(continued)

PROBLEMS

Say that NASA is launching two small communications 

satellites to evaluate different methods of sending data 

home to Earth. The optical transmitter satellite is 41 cm 

by 31 cm by 21 cm. The RF transmitter satellite is 35 cm 

by 48 cm by 22 cm. The experiment requires two different 

rocket flights, one for the RF payload and one for the 

optical payload.

1. Sketch a diagram of each box-shaped payload. 

2. Our two communications satellites have a few 

rockets to pick from. Larger rockets tend to be more 

expensive, so we want to use the smallest possible 

rocket that will still fit our payload. First, let's consider 

an easy way to simplify: When placing each payload 

inside the rocket fairing, which face should be oriented 

toward the base of the rocket so that it fits in the 

smallest space? For this exercise, assume that NASA 

can manufacture fairings as tall as needed, so height is 

not a constraint. Explain your reasoning.

Fairing splits and drifts away 

from the satellite

Payload 

fairing

Payload 

fairing

Payload 

envelope

Satellite

Satellite

4.5 m 

diameter

8.6 m 

height

Upper 

Stage Upper 

Stage

Currently, NASA uses radio frequency (RF) 

transmitters, but is also developing optical 

communications technologies that use infrared 

lasers to send more data than traditional RF. 

Optical terminals are also smaller and lighter than 

comparable RF dishes, which can leave more room 

on the spacecraft for science instruments.

The National Aeronautics and Space 

Administration (NASA) reaches for 

new heights and reveals the unknown 

for the benefit of humankind. To 

accomplish this objective, NASA 

launches spacecraft with science 

instruments that can study the universe. 

During launch and operations, NASA's 

space communications and navigation 

infrastructure enables the spacecraft 

to communicate to operators on Earth. 

The rockets that launch these missions 

must be streamlined—built smooth 

and tapered—so that they are as 

aerodynamic as possible.

Often, NASA must construct a shell, 

called a fairing, to fit over the spacecraft 

(and sometimes the upper stage) at the very tip of the 

rocket. The fairing protects the spacecraft during launch 

and keeps the rocket aerodynamic. Once the rocket 

has left the atmosphere, the fairing can be ejected, or 

jettisoned, and either discarded or, sometimes, recovered.

Payload designers—the engineers who build spacecraft 

and satellites—need to make sure that spacecraft can 

fit inside a rocket’s fairing. While many payloads are 

carefully folded or hinged to fit inside, in this exercise you 

will practice with simpler boxes.
PAGE 2

Geometry of Circles

PROBLEMS (continued)

3. Now we want to determine the smallest fairing

each of our two payloads can use. This is

complicated by the fact that the outer diameter of

the rocket fairing does not represent the volume

we have to work with; we must account for the

thickness of the fairing shell. What is the smallest

usable fairing for each payload?

Consider a rectangle centered on the circles’ centers.

How will the length and width of the rectangle relate to

the diameter of the circles? Remember the Pythagorean

theorem, where a right triangle’s hypotenuse,

c = √(a2 + b2) with a and b being the lengths of the other

two sides.

Inner Diameter

Fairing

Thickness

Outer 

Diameter

Rocket type Fairing outer 

diameter

Fairing 

thickness

Rocket A 0.26 m 2 cm

Rocket B 0.44 m 3 cm

Rocket C 0.50 m 4 cm

Rocket D 0.76 m 8 cm

a. Assuming that we center the payload in the fairing,

determine the largest radius of the payload’s footprint.

b. Determine the largest radius that would fit inside

each of our four rocket fairing options. Think carefully

about how you factor the fairing thickness into your

equation!

c. Convert your payload radii and rocket fairing radii to

common units.

d. Compare each of the rocket fairing radii to the

payload radii. Which payloads can fit in which

rockets? What’s the least expensive (smallest) rocket

we can use for each of our two payloads?

NASA’s Laser

Communications Relay 

Demonstration (LCRD)

is the next step in optical 

communications.

Hosted on a U.S. Space Force spacecraft, 

LCRD will demonstrate optical technology 

in geosynchronous orbit 22,300 miles above 

Earth’s surface. LCRD is built and managed 

by NASA’s Goddard Space Flight Center in 

Greenbelt, Maryland.

LCRD is a technology demonstration that will 

pave the way for future optical communications missions. When NASA has developed 

a new way to solve a problem and wants 

to show how it works, they create a technology demonstration. LCRD’s first orbiting 

experimental user will be the International 

Space Station's Integrated LCRD Low-Earth 

Orbit User Modem and Amplifier Terminal 

(ILLUMA-T). The terminal will receive science data from experiments and instruments 

aboard the space station and then transfer 

the data to LCRD, which will then transmit 

them to a ground station. After the data arrive 

on Earth, they will be delivered to operation 

centers and scientists to help make decisions 

about the mission.

LEARN MORE

Learn more about LCRD’s predecessor, 

the Lunar Laser Communications 

Demonstration (LLCD), at 

http://go.nasa.gov/LLCD_history.

Learn more about LCRD at 

http://go.nasa.gov/LCRD_overview.