ArticlePDF Available

## Abstract and Figures

Although they are sometimes considered problematic to grasp by students, the concepts behind non-Euclidean geometry can be taught using astronomical images. By using photographs of the Moon taken with a smartphone through a simple telescope, we were able to introduce these concepts to high-school students and college newcomers. By recognizing different Moon geological structures within the photograph, we teach students how to calculate distances of mountain ranges or areas of craters on the Moon's surface, introducing the notions of geodesics and spherical triangles. Furthermore, students can empirically see that the correct estimations for the actual values cannot be obtained using flat geometry. Instead, by using three-dimensional curved geometry, precise estimates of lengths and areas of geological elements in the Moon can be computed with less than 4% error. These procedures help students understand, concretely, non-Euclidean geometry concepts.
Content may be subject to copyright.
Using smartphone photographs of the Moon to acquaint students
with non–Euclidean geometry
Hugo Caerols,1, 2, 3, Rodrigo A. Carrasco,1, 4, and Felipe A. Asenjo1,
3IAU National Astronomy Education Coordinator (NAEC) Chile team.
4Data Observatory Foundation, Santiago 7941169, Chile.
(Dated: August 19, 2021)
Non-Euclidean geometry can be taught to students using astronomical images. By using pho-
tographs of the Moon taken with a smartphone through a simple telescope, we were able to in-
troduce these concepts to high-school students and lower-level college students. We teach students
how to calculate lengths of mountain ranges or areas of craters on the Moon’s surface and introduce
ideas of geodesics and spherical triangles. Students can see that accurate measurements cannot be
obtained using ﬂat geometry. Instead, by using three–dimensional curved geometry, estimates of
lengths and areas can be computed with less than 4% error.
I. INTRODUCTION
In classical astronomy and navigation, consideration
of curved spaces and surfaces is essential, and now the
entire Universe is recognized as being non-Euclidean.
However, introducing students to the geometry of curved
surfaces is a challenge.
The use of astronomy-related activities and evidence-
based active engagement have been shown to signiﬁcantly
help students grasp complex concepts, encourage them
to pursue science studies, and reduce the gender gap
in STEM areas.[1012,14,19] This improvement is fur-
ther enhanced by having students use real data which
they acquire themselves.[1,7] Hence, what better way
to introduce the diﬀerences between Euclidean and non-
Euclidean geometries than with the Moon’s spherical ge-
ometry?
In this work, we demonstrate the use of low-resolution
photographs of the Moon taken with a smartphone and
a simple telescope to acquaint students with concepts of
three–dimensional curved geometry such as distances and
areas, and, at the same time, introduce concepts such
as geodesics. Any student interested in how geological
features on the Moon can be measured can perform their
own calculations with these methods.
We start in Section II by showing that our natural
satellite presents a valuable teaching tool to introduce
curved geometry. In Section III we show that estimates
of distances in craters or valleys on the Moon based on
proportionality arguments are valid only for lengths small
compared to the Moon’s radius. In Section IV we explain
how the methods of non–Euclidean geometry are helpful
for estimating the lengths of mountain chains; we also
show how the results diﬀer when considering a ﬂat geom-
etry. In the last part of this work we use those methods
to calculate the area of a lunar mare by using a sum of
simple polygons. To perform this task, we introduce in
Section Vconcepts of spherical trigonometry, which al-
low us to obtain any desired area as the sum of the areas
of spherical triangles.
II. PHOTOGRAPHING THE MOON
Earth is unique in the Solar System in having a moon
which is large in comparison to the size of its parent
planet. Furthermore, since we always see the same face
of the Moon due to tidal locking, viewers can see and
photograph the same features in any season.
In principle, any photograph(s) of the Moon can be
used. However, it is important that students participate
in this activity by taking their own photographs. The
Moon passes through a cycle of phases during a month,
which results in varying apparent luminosity. To obtain
the best photographs for these activities, we recommend
against using photos taken when the Moon is full. At full
phase the light reﬂected to Earth is perpendicular to the
Moon’s surface, which tends to ﬂatten the appearance of
features and prevents observation of speciﬁc details. We
suggest planning the observations to take photographs
within a few days of ﬁrst-quarter phase, when the Sun’s
rays highlight the moon’s features. Additionally, the ﬁrst
quarter Moon is always visible in the hours just after
sunset, which is a convenient time.
An example of a good photograph of the Moon is shown
in Fig. 1. The numbered craters in this ﬁgure are used in
the following sections, and they are shown in more detail
in Fig. 2. This was taken using an iPhone 6 attached to
a telescope with a NexYZ 3-Axis Universal Smartphone
adapter. We used a Celestron Nexstar 8SE telescope,
which has a focal length of 2,032 mm and an ocular with
a focal length of 25 mm; this combination gives a magni-
ﬁcation of 80. We also used a standard lunar ﬁlter to
cut down the Moon’s glare. A smaller and cheaper tele-
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
2
scope can certainly be used to obtain the desired images.
After the photograph is taken, a smartphone app such
as Snapseed can be used to enhance the contrast in the
picture.
Although the orientation of the photograph is not es-
sential for the proposed calculations, it should be noted
that telescopes invert images and that they need to be
reversed or rotated to represent what is seen with the
naked eye. Furthermore, images of the Moon taken in the
northern and southern hemispheres are reversed. This
last point is relevant in that the image of the Moon in
Fig. 1was taken from the southern hemisphere.
III. SMALL DISTANCES, FLAT GEOMETRY,
AND PROPORTIONALITY
For distances on the Moon’s surface that are small in
comparison to its radius, Euclidean geometry is suﬃcient
to accurately estimate the distance between two points
that are not near the limb. Near the limb, Euclidean
geometry will fail, as we will see in the case of Plato
FIG. 1: Moon right after ﬁrst quarter, 9 days old and
with a luminosity of 67.4 of that of the full Moon.
Photograph taken with a smartphone and telescope on
April 21, 2021 at 20:27, from Santiago, Chile. The
numbers indicate craters and mare that we analyze in
this study.
Crater dCp cm dCkm dCexact km error %
(1) Plato 0.3212 104.61 100.68 3.91%
(2) Aristillus 0.1653 53.84 54.37 0.98%
(3) Autolycus 0.1197 38.99 38.88 0.27%
(4) Archimedes 0.2476 80.64 81.04 0.49%
(5) Timocharis 0.1032 33.61 34.14 1.55%
(6) Eratosthenes 0.1757 57.22 58.77 2.63%
(7) Copernicus 0.2893 94.22 96.07 1.92%
(8) Eudoxus 0.1445 47.07 67.00 29.7%
TABLE I: Diameters for the seven craters of Fig. 2,
crater.
Figure 2shows a magniﬁcation of the lower part of
Fig. 1. We can use proportionality to calculate the diam-
eter of these eight craters. The real values of the craters’
diameters dCare small compared to the diameter of the
Moon 2RM, where the Moon’s radius is RM= 1,737.4
km. In the Euclidean approximation, the proportions
between the distances are linear. Thus, a diameter of a
crater can be computed as
dC=RM
RMp
dCp ,(1)
where RMp is the Moon’s radius in the photograph and
dCp is the crater’s diameter in the photograph. To
obtain those values, we use GeoGebra to analyze the
photographs. With this software we can calculate how
large the diameter of the craters and the Moon’s radius
are in the photograph on a centimeter scale. A detailed
procedure to obtain those values can be found in Ref.
. To measure the diameter of a crater in GeoGebra,
place two points that are diametrically opposite at the
crater’s edge. This choice should be made as precisely
as possible, using the zoom-in facility. This pro cedure
helps reduce the errors in the measurements done in the
software.
Applying the above procedure to Figure 2results in
RMp = 5.3345 cm. The same can be performed to ob-
tain the diameters dCp of diﬀerent craters (listed in Table
I). Thus, through this procedure, any crater’s diameter
dCcan be obtained by measuring its equivalent diam-
eter in the photograph and multiplying the result by
RM/RMp = 325.6913 km/cm. In Table Iwe show the
eight diﬀerent values dC p from the photograph (in cm)
and the calculated diameters dC(in km) obtained from
Eq. (1).
We compare the estimated diameters with the exact
values dCexact obtained from the Working Group for
Planetary System Nomenclature (WGPSN) from the In-
ternational Astronomical Union (IAU). The error be-
tween the actual value of the crater’s diameter and the
estimate done through our linear calculation is computed
as 100|dCexact dC|/dC exact.
Clearly, the best results are obtained for the three
smaller craters at the center of the photograph. Con-
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
3
FIG. 2: Zoomed in view of Fig. 1. Eight craters with
their respective names: (1) Plato, (2) Aristillus, (3)
Autolycus, (4) Archimedes, (5) Timocharis, (6)
Eratosthenes, (7) Copernicus, (8) Eudoxus. The straight
line between points Aand Brepresents the
approximated Euclidean ﬂat distance of Montes
Apenninus. The straight line between points Cand D
represent the approximated Euclidean ﬂat distance of
Montes Caucasus.
versely, the most signiﬁcant error (4%) occurs for the
Plato crater. This error can be partially attributed to
the crater’s size and the lack of good luminosity at its
position close to the edge of the photo. However, this er-
ror is an indicator that the linear Euclidean approxima-
tion for distances of this scale is already breaking down.
For such cases we need to consider the curvature of the
Moon’s surface and use non–Euclidean geometry.
The size of the pixels in the photo determines the
uncertainty of the measurements. Here, this measures
10.6/2448 = 0.00433 centimeters (2448 is the width of
the image in pixels). When scaled to the size of the
Moon, this value gives between 1 and 2 km per pixel.
As we will see in the following, this depends on whether
the crater is near the center or closer to the edge of the
photo. The errors obtained in Tables I and II are in this
range.
IV. LARGE DISTANCES, CURVED
GEOMETRY AND GEODESICS
For greater distances on a curved surface, using pro-
portionality is insuﬃcient to provide estimates with rea-
sonable accuracy, as shown in the previous section. To
calculate curved-surface distances it is essential to intro-
duce the concept of geodesic.
A geodesic is the line of minimum length between two
points on a surface. This can involve any geometry, ﬂat
or curved, and, therefore, a geodesic can be straight or
curved. For example, given two points in a plane, the
geodesic is the straight line joining the points. But for
the curved surface of a sphere, the geodesic is an arc of
a great circle which passes through the two points and
whose center lies at the center of the sphere.
For measuring large distances between two points on
the Moon’s surface, it is suﬃcient to approximate the
Moon as a sphere and calculate the geodesics accordingly.
In what follows, we consider the actual values of lunar
features to be those appearing in Google Earth Pro free
software. 
As an example, we measure the length of the Montes
Apenninus mountain chain; see Fig. 2. The distance be-
tween points Aand Bon considering the surface to be
Euclidean and using the proportionality approach of the
preceding section is 552 km, but the real distance be-
tween these points is 600 km, an error of 8 %. On con-
sidering two points even further apart, the error grows
signiﬁcantly. For example, the correct distance between
craters (1) and (7) is 1,300 km, but using proportionality
gives only 1,081 km, an error of about 17%.
To consider non-Euclidean geometry, we ﬁrst note that
surface points on any photograph of the Moon project
into the plane of the sky. This plane, which will in-
clude the Moon’s polar axis, will divide the Mo on into
two. This is illustrated in Fig. 3, where the circle repre-
sents the projection of the Moon onto this plane. Con-
sequently, actual geodesic distances between two points
in a photograph must be calculated by “back-projecting”
them onto the spherical surface.
In Fig. 3,Pand Qrepresent two points in the pho-
tograph of the Moon, Pand Qare the corresponding
points projected onto the Moon’s surface, and CMis the
center of the Moon. The line P′′Qis a copy of the line
P Q, but shifted toward the lunar surface; that is, to the
outside observer, P′′Qoverlays P Q. The desired dis-
tance is the length of the curved geodesic between points
Pand Q. This can be calculated using the Pythagorean
theorem and trigonometry as follows:
Geodesic
z}|{
PQ=αRM,(2)
where αis the angle formed by the three points P,CM,
and Q,
α=
\
PCMQ= 2 arcsin d(P, Q)
2RM,(3)
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
4
FIG. 3: Scheme where Pand Qrepresent two points in
the Moon photograph. Points Pand Qcorrespond to
their projections in the Moon’s spherical surface. Other
points are depicted in order to perform the calculation.
where d(P, Q) is the straight-line distance between
points Pand Q. Hereafter, d(A, B ) is used to denoted
the straight-line distance between any points Aand B.
This distance can be obtained by the Pythagorean theo-
rem as
d(P, Q) = pd(P′′, Q)2+d(P′′ , P )2,
=pd(P, Q)2+d(P′′, P )2,(4)
where we have used d(P′′, Q) = d(P, Q). Now,
d(P′′, P ) = d(P , P )d(Q, Q),(5)
and
d(P, P ) = qR2
Md(CM, P )2,
d(Q, Q) = qR2
Md(CM, Q)2.(6)
Using equation (2), we can obtain the real distance
between two points on the Moon’s surface by using the
linear distance between those two points in the ﬂat pho-
tograph and their distances (again in the photograph) to
the center of the Moon by using the following expression:
geodesic
z}|{
PQ= 2RMarcsin
1
2RMsd(P, Q)2+qR2
Md(CM, P )2qR2
Md(CM, Q)22
.(7)
This result can be programmed in GeoGebra to eﬃciently compute estimates for several structures on the Moon’s
surface. We have left a sample version in , which can be used if the calculations are too advanced for a group of
students. As an application, we use equation (7) to estimate real distances in Fig. 2for several points and compare
them with the values found in Google Earth Pro software .
In Table (II) we summarize the sizes (in km) of several structures and percent errors. We show estimates obtained
considering a ﬂat model, a curved model (computed with equation (7)), and the distance obtained in Google Earth
Pro; the errors are calculated using the values obtained from ref . These values are obtained from measurements
of satellite photographs taken by NASA and partner agencies.
The ﬂat approximation produces errors of 10% or larger, whereas those from the the curved approach are signiﬁcantly
smaller. This error is more signiﬁcant if the feature considered is closer to the Moon’s perimeter in the photograph,
as is the case for the crater Eudoxus and Mare Crisium.
As shown in Table (II), the ﬂat estimate is very poor when this occurs. Like before, there are an uncertainty in
these measurements of 1-2 km per pixel due to the pixel size. This uncertainty is, however, much smaller than the
improvement achieved in the estimates when considering a curved surface over a ﬂat one, which is the central point
of this activity.
V. CALCULATING AREAS AND THE
SPHERICAL TRIANGLE
Now that students have been introduced to the con-
cepts of curved surfaces and how to compute distances
over them, we proceed to calculate the area of a curved-
surface feature using a ﬂat Moon photograph. We apply
this later to obtain the area of a polygonal region of the
Moon, Mare Crisium. But ﬁrst we need to describe some
results from spherical trigonometry.
This ﬁgure shows three points on a sphere of radius r, to-
gether with the arcs of the great circles that pass through
them. When the angles α,β, and γare known, the area
can be calculated straightforwardly from the expression

A(ABC) = (α+β+γπ)r2.(8)
This result is known as Girard’s Theorem; it also indi-
cates that the sum of interior angles for a spherical tri-
angle is always greater than πradians, something that
does not occur in Euclidean geometry.
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
5
FIG. 4: Spherical triangle ABC in a spherical surface
and its interior angles 
To calculate these angles, we use the cosine rule for the
sides of a spherical triangle, which relates the angles to
the side lengths of the triangle. In Figure 4, let ABC
be the spherical triangle with sides AB,B C, and CA,
which have geodesic lengths c,a, and brespectively. The
angles at A,B, and Care α,β, and γ, respectively. The
cosine law gives
α= arccos cos ˜acos ˜
bcos ˜c
sin ˜
bsin ˜c!,
β= arccos cos ˜
bcos ˜acos ˜c
sin ˜asin ˜c!,
γ= arccos cos ˜ccos ˜acos ˜
b
sin ˜asin ˜
b!.(9)
where ˜a=a/r,˜
b=b/r, and ˜c=c/r. The lengths a,b,
and ccan be calculated with the procedures developed in
Sec. IV, and then, using Eqs. (9), we obtain the angles.
With this information, we can calculate the required area
using Eq. (8).
VI. ESTIMATING AREAS ON THE SURFACE
OF THE MOON USING SPHERICAL
TRIANGLES AND POLYGONS
We are now in a position to compute an area on
the Moon’s surface using the methods from the previ-
ous sections. The previous formulas can be programmed
into GeoGebra to compute the estimates. We provide
sample images and GeoGebra ﬁles in a public reposi-
tory that can be found at https://github.com/raxlab/
moonGeometry. Using this code, one can give three
points in the photograph, and, by using a spreadsheet,
compute the total curved area of interest. Any area can
FIG. 5: (a) Zoomed in view of photograph of Fig. 2
showing the spherical triangle with vertices in the
center of craters Aristillus,Autolycus, and Archimedes.
Results obtained with the GeoGebra code are shown.
(b) A spherical triangle is formed by the same craters
constructed with Google Earth Pro software. The
results for the curved areas using the software and our
calculations are displayed in both ﬁgures for
comparison.
be approximated as a sum of several spherical-triangle
areas. Therefore, the key is to estimate as accurately as
possible the area of each spherical triangle. An example
of the result is given in Fig. 5.
A screenshot of the GeoGebra code is shown in Fig-
ure 5(a). We calculate the area of the spherical triangle
between the centers of the craters Aristillus, Autolycus,
and Archimedes. These correspond to the points F, G,
and H in the GeoGebra code, respectively, and to points
2, 3, and 4 in Figure 2. Part of the code is depicted on the
right-hand side of Figure 5(a). We show the curved dis-
tance between points F and G, which is equal to 141.72
km, and the angles in each vertex, in radians.
The normalized curved area is also shown, which, when
multiplied by the Moon’s radius (Eq. (8)), results in the
area for the spherical triangle, which emerges as 7,135.35
km2. As a comparison, the estimation using the ﬂat
area calculation gives only 5,845.25 km2. To compare
these estimates, we use the Google Earth Pro software
to calculate the same spherical triangle. This pro cedure
is shown in Figure 5(b), and results in a total area of
7,119.2 km2. The estimate using curved areas obtained
with our low-resolution photograph diﬀers by only 0.2%
from the result using satellite photographs from Google
Earth Pro.
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
6
Finally, the same two ﬁgures show how the curvature
aﬀects the distances. In Figure 5(a), the perimeter of the
triangle calculated by the sum of the three geodesics of
each side (Eq. (7)) gives 429.12 km. The same perime-
ter using Google Earth Pro results in 429.16 km. These
two results are virtually identical (0.009% diﬀerence),
which illustrates how good the calculation of curved ge-
ometrical structures can be when using a low-resolution
photograph and care is taken in specifying the points.
We can use the above spherical triangle approach to
obtain areas from the largest observable Moon structures,
the lunar maria. In particular, we analyze Mare Crisium,
which is shown as element 9 in Figure 1. In Figure 6(a)
we present a magniﬁed view.
The procedure is as follows. We can divide an area
into several spherical triangles; the total area will be the
sum of all the individual areas. This is shown in Figure
6(a). This image shows how we calculate the total area
by estimating the mare’s inner boundary with an inte-
rior triangulation, dividing it into ten triangles. Figure
6(b) shows the same procedure for the outer boundary
with an exterior triangulation; we do this to compute
bounds on the estimate of the total area. Deﬁning the
correct boundaries in these types of photographs is very
challenging.
Using this procedure, we obtain a lower bound for the
total area (the inner boundary estimate) of 133,993 km2.
In comparison, the outer-boundary estimate is 205,684
km2. The average of these is 169,839 km2, which diﬀers
by only about 3.5% from the documented area of 176,000
km2. If we calculate these areas using a proportional
approximation of the ﬂat areas, we obtain lower and up-
per bounds of 48,497 and 70,551 km2; both of these are
very diﬀerent than the documented area.
Figure 6(c) shows the same calculation performed with
Google Earth Pro for the inner boundary. This gives a
value of 138,012 km2, very close (3% diﬀerence) to
our estimate. Furthermore, notice that in Figure 6(a),
the distance between points Vand Qseems shorter than
distance between Nand T. That the opposite is actually
the case is shown in Figure 6(c). Our calculation using
curved distances automatically deals with this.
In the above analysis for Mare Crisium, the inner and
outer boundaries result in estimates that diﬀer by 71,691
km2(about 34% of the outer-boundary estimate). It is
important to identify the origin of this variation. To see
and comprehend an image, humans need between approx-
imately 103and 104pixels.[5,18] With the same proce-
dure used to measure the areas in Fig. 1, we can measure
the equivalent area that corresponds to one pixel. We
ﬁnd that each pixel corresponds to roughly between 3
and 8 km2. This implies that the minimum error we
can achieve in an area estimate will be of the order of
105km2, which is of the same order as those obtained
for Mare Crisium. Although the errors and uncertainties
can be improved by choosing a better point to estimate
the triangulation or by increasing the resolution of the
captured image, there is always error associated with the
FIG. 6: Mare Crisium. (a) Interior triangulation to
calculate its area, by using a ten-sided polygon. (b)
Exterior triangulation with a ten-sided polygon. (c)
Interior triangulation of the area, by using a ten side
polygon, but with Google Earth Pro software.
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
7
pixel sizes of the adopted image.
VII. CONCLUSIONS
This work shows how Moon photography can be used
to teach non-Euclidean geometry. Although this sub-
ject can be complicated, we believe that by using their
own photographs of the Moon, students can better un-
derstand the importance of curved geometry and gain a
deeper appreciation that celestial objects are not ﬂat.
The activity starts with a photograph of the Moon
taken by students themselves with a smartphone. When
distances and areas of some of the structures are calcu-
lated over the picture; for example, for the lengths of
mountain chains or the sizes of craters, students will re-
alize that the results of ﬂat calculations can diﬀer enor-
mously from the documented values. This insight is the
key to introduce curved geometry, which can be devel-
oped using the same photographs and relatively sim-
ple mathematics. In measuring distances, students learn
about geodesics on the surface of a sphere and how to use
them. We further extended the study to calculate non-
ﬂat areas, for which the concepts of spherical trigonom-
etry are essential.
The activity can also include creating a spreadsheet in
GeoGebra. This free software can automate the proce-
dure of using the points needed to construct distances
and triangles. Students will come to realize that their
decisions regarding selected points will impact their ﬁ-
nal estimates lengths and areas. This diﬃculty will put
the process of measuring geological formations in other
moons or planets into context.
As an initial complementary activity, students can per-
form the same analysis for distances and areas for a pho-
tograph of a soccer ball or basketball. This has the
additional pedagogical beneﬁt that they can compare
their calculations with actual lengths measurements, per-
formed by using a string or ﬂexible tape between two
points. This can be considered as an engaging activity.
Again, students will ﬁnd that to get reasonable estimates
they have to consider the curvature of the surface. If they
can program their spreadsheets and secure measurements
for a ball, they should be able to do it for the Moon as
well.
We have used parts of the approach described here
in courses for high school mathematics teachers, who be-
came motivated to adopt these exercises. We believe that
the best way to use this approach is by challenging teams
of learners. Depending on the context, the teacher can se-
lect either an active expository or a dynamic class, where
the students can use the above equations as an input to
perform their calculations and predictions. In the lat-
ter, teams of students are set to work consecutively the
ology used to address the problem, each topic can be
covered in one or two 60-minute classes. For college-level
courses, the teacher can do a guided activity as described
above. In contrast, for summer workshops in astronomy,
the teacher can do a research activity where the professor
can exhibit the main questions or challenges of measuring
distances and areas on spherical objects and then chal-
lenge students to do so with a photo of the Moon, giving
In summary, we believe that the approach to intro-
ducing curved geometry outlined here is challenging and
motivating for participants. Our experience has been pre-
cisely this for both students and teachers.
An extended version of this paper can be found in .
Acknowledgments
This research was partially funded by grants
ASTRO020-0018, ASTRO20-0058, and FONDECYT
1180139 from ANID, Chile. The authors would also like
to thank the reviewers and the editors for their comments
and recommendations, which improved our work.
 Afrodita L. Boldea and Ovidiu Vaduvescu. Teaching stu-
dents about informatics and astronomy using real data
for detection of asteroids. European Journal of Physics,
38(5):55706, Aug 2017.
 David A. Brannan. Geometry. Cambridge University
Press, 2nd edition, 2011.
 Hugo Caerols and Felipe A. Asenjo. Estimating the
Moon-to-Earth Radius Ratio with a Smartphone, a Tele-
scope, and an Eclipse. The Physics Teacher, 58(7):497–
501, 2020.
 Rodrigo A. Carrasco, Hugo Caerols, and Felipe A.
Asenjo. Moon Geometry Repository v.1.3.https://
github.com/raxlab/moonGeometry, 2021.
 Yang Cai. How Many Pixels Do We Need to See Things?
In Sloot P.M.A., Abramson D., Bogdanov A.V., Gor-
bachev Y.E., Dongarra J.J., and Zomaya A.Y., editors,
Lecture Notes in Computer Science, ICCS 2003, pages
1064–1073. Springer Berlin Heidelberg, 2003.
 Bonaventura Cavalieri. Directorium generale uranomet-
ricum. 1632.
 Benjamin Farr, GionMatthias Schelbert, and Laura
Trouille. Gravitational wave science in the high school
classroom. American Journal of Physics, 80(10):898–904,
2012.
 GeoGebra. GeoGebra Classic 5.0 User Manual, 2019.
https://www.geogebra.org/, 2021.
 Google. Go ogle Earth Pro User Manual, 2020. https:
 Christian R. Hayes, Allison M. Matthews, Yiqing Song,
Sean T., Linden, Robert F. Wilson, Molly Finn, Xi-
aoshan Huang, Kelsey E. Johnson, Anne M. McAlis-
ter, Brian Prager, Richard Seifert, Sandra E. Liss, An-
drew M. Burkhardt, and Nicholas Troup. First results
from the Dark Skies, Bright Kids astronomy club draw-
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
8
a-scientist test. Physical Review Physics Education Re-
search, 16(1):10131, may 2020.
 Naﬁs I. Karim, Alexandru Maries, and Chandralekha
Singh. Do evidence-based active-engagement courses re-
duce the gender gap in introductory physics? European
Journal of Physics, 39(2), 2018.
 Wei Lee, Heidi L. Gilley, and Joshua B. Caris. Finding
the surface temperature of the Sun using a parked car.
American Journal of Physics, 65(11):1105–1109, 1997.
 Gazetteer of Planetary Nomenclature. International As-
tronomical Union (IAU) Working Group for Planetary
System Nomenclature (WGPSN), 2020.
 Luisa M. Rebull, Debbie A. French, Wendi Lau-
rence, Tracyanne Roberts, Michael T. Fitzgerald, Varou-
jan Gorjian, and Gordon K. Squires. Major outcomes
of an authentic astronomy research experience profes-
sional development program: An analysis of 8 years of
data from a teacher research program. Physical Review
Physics Education Research, 14(2):20102, jul 2018.
 William M. Smart and Robin M. Green. Textbook on
spherical astronomy. Cambridge University Press, 1977.
 Snapseed. Snapseed, 2020.
 Isaac Todhunter. Spherical Trigonometry. 1886.
 Antonio Torralba. How many pixels make an image?
Visual Neuroscience, 26(1):123–131, jan 2009.
 Malcolm Wells, David Hestenes, and Gregg Swackhamer.
A modeling method for high school physics instruction.
American Journal of Physics, 63(7):606–619, 1995.
 James L. Whitford-Stark. A preliminary analysis of lu-
nar extra-mare basalts: Distribution, compositions, ages,
volumes, and eruption styles. The moon and the planets,
26(3):323–338, 1982.
 John Wilkinson. The Moon in Close-up: A Next Gener-
ation Astronomer’s Guide. Springer Science & Business
Media, 2010.
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
9
Elements (A) (B) (C) (D) (E)
straight-line geodesic accepted % errors % errors
Eudoxus, point 8 in Figs. 1and 247 68 67 1.49 30.9
Montes Caucasus 327 438 443 1.13 25.3
Mare Crisium, point 9 in Fig. 1203 546 555 1.62 62.8
Montes Apenninus 552 596 600 0.67 7.4
Distance between points 1 and 7 in Fig. 21,081 1,302 1,300 0.15 17
TABLE II: Distances (in km) on the Moon’s surface. Column (A) lists lineal (straight-line) distances, column (B)
are geodesic distances based on Eq. (7), column (C) are Google Earth Pro “accepted” values, column (D) are the %
errors between our estimates and the accepted values, and column (E) are the % errors between the straight-line
and geodesic distances. Mare Crisium has an elliptical shape. We use the distance between points Vand Qin ﬁgure
6(a) to calculate its radius, as explained later.
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.