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National Aeronautics and 

Space Administration

NASA Space Communications 

and Navigation

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

Geometry of Circles GRADES 9–10

PAGE 1

(continued)

Answer Key

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

Using box-shaped payloads might not fill the volume efficiently,

but the shape has been picked for simplicity of the exercise.

48

22

35

41

21

31

Optical transmitter

RF transmitter

2. 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?

Select the smallest face (just pick the two smallest dimensions

from among the supplied length/width/heights).

RF: 22 < 35 < 48 Optical: 21 < 31 < 41

22

35

21

31

3. What is the smallest fairing each of our payloads can use?

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

determine the largest radius of the payload’s

footprint.

The largest radius of the payload is the hypotenuse r:

Fairing

Thickness

Fairing

Outer 

Diameter

q

r

r

l

w

w

2

w

2

l

2

l

2

r2 = ∙

w

2 ∙

2

+ ∙

l

2 ∙

2

r = ∙∙

w

2 ∙

2

 + ∙

l

2 ∙

2

Evaluate: Is r < q?

Optical Transmitter:

rOpt = ∙∙

21cm

2 ∙

2

+ ∙

31cm

2 ∙

2

rOpt = ∙(10.5 cm)2

 + (15.5 cm)2

rOpt = ∙110.25 cm2

 + 240.25 cm2

rOpt = ∙350.5 cm2

rOpt = 18.7 cm

INSTRUCTOR’S GUIDE

RF Transmitter:

rRF = ∙∙

22cm

2 ∙

2

+ ∙

35cm

2 ∙

2

rRF = ∙(11 cm)2

 + (17.5 cm)2

rRF = ∙121 cm2

 + 306.25 cm2

rRF = ∙427.25 cm2

rRF = 20.7 cm

To see an example of how NASA folds spacecraft to 

fit inside rocket fairings, animations of the Tracking 

and Data Relay Satellite, known as TDRS-K, can be 

seen here: http://go.nasa.gov/TDRS-K_videos. For 

an animation of the James Webb Space Telescope 

(JWST) deployment, visit: http://go.nasa.gov/

JWST_deployment

Additional JWST educational activities and resources 

can be found at http://go.nasa.gov/JWST_educators_

formal and http://go.nasa.gov/JWST_activities.

q = 2

Fairing Outer 

Diameter

Fairing ∙ ∙ ∙Thickness∙ – 2 ×
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Geometry of Circles

b. Determine the largest radius that would fit inside 

each of our four rocket fairing options.

Max. diameter (p) = Fairing outer 

diameter – (2 × Fairing 

thickness )

pA = 0.26 m – 2(0.02 m) = 0.22 m

pB = 0.44 m – 2(0.03 m) = 0.38 m

pC = 0.50 m – 2(0.04 m) = 0.42 m

pD = 0.76 m – 2(0.08 m) = 0.60 m

Max. radius (q) = p

2

qA = 0.11 m qC = 0.21 m

qB = 0.19 m qD = 0.30 m

c. Convert your payload radii and rocket fairing radii to 

common units.

We’re about ready to make the crucial comparisons here, but 

our payload radii and our fairing radii are in different units (m 

and cm). Get both measurements in the same unit. We’ll use 

cm here, but either is valid:

qA = 0.11 m × (

100 cm 

m ) = 11.0 cm qC = 21.0 cm

qB = 19.0 cm qD = 30.0 cm

d. Compare 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?

Which fairings have a largest possible radius (q) that’s larger 

than the largest radius (r) of the payload?

Optical Transmitter:

q r

Rocket A: Is 11.0 cm ≥ 18.7 cm? No. It doesn’t fit.

Rocket B: Is 19.0 cm ≥ 18.7 cm? Yes. It fits.

Rocket C: Is 21.0 cm ≥ 18.7 cm? Yes. It fits.

Rocket D: Is 30.0 cm ≥ 18.7 cm? Yes. It fits.

RF Transmitter:

q r

Rocket A: Is 11.0 cm ≥ 20.7 cm? No. It doesn’t fit.

Rocket B: Is 19.0 cm ≥ 20.7 cm? No. It doesn’t fit.

Rocket C: Is 21.0 cm ≥ 20.7 cm? Yes. It fits.

Rocket D: Is 30.0 cm ≥ 20.7 cm? Yes. It fits.

While laser communications can provide more 

information at once to scientists, clouds can 

disrupt laser signals as they enter Earth’s 

atmosphere. LCRD sends its information to 

two ground stations located on Table Mountain 

in California, and on Haleakala– Volcano on 

the island of Maui, Hawaii. These remote and 

high-altitude locations were chosen for their 

clear weather conditions. 

NOTE: The rocket dimensions selected for this exercise 

are based on real sounding rockets used by NASA. 

Sounding rockets are smaller vehicles and make suborbital 

flights: they do not have the power to make it into orbit. 

These rockets fall back to Earth on parachutes after their 

fuel is spent. Learn more about sounding rockets at 

http://go.nasa.gov/WFF_soundingrockets. 

The payload and fairing volumes for this exercise are 

loosely based on real transmitters but were otherwise 

selected for mathematical convenience.