Page 1 of 10
Transactions on Engineering and Computing Sciences - Vol. 13, No. 1
Publication Date: February 25, 2025
DOI:10.14738/tecs.131.18307.
Abunaieb, S. (2025). The Sun and Moon are Larger by About 1.4% than we Think. Transactions on Engineering and Computing
Sciences, 13(1). 147-156.
Services for Science and Education – United Kingdom
The Sun and Moon are Larger by About 1.4% than we Think
Salah Abunaieb
Sogex Oman LLC, Muscat, Sultanate of Oman
ABSTRACT
This is a simple geometric modification to the currently recorded sizes of the sun
and moon. The concept is based on the well-known observation that when looking
at an object, it appears to progressively taper off toward its distant end. Spherical
objects are no exception, even though their apparent tapering is not readily
noticeable due to their unique geometry. As shown in this study, when looking at
the solar disc or at the moon, we are actually looking at the base side of a seemingly
slightly egg-shaped sun or moon. Given that the sun and moon are spherical in
reality, the geometrically adjusted sizes show that each is about 1.4% bigger in
volume than currently thought.
Keywords: Sun, Moon, Angular Size, Solar Disc, Face of the Moon, Apparent Tapering,
Small Angle, 3D Objects, Oval, Egg-shaped, Depth Perception, Sphere, Relative Size.
INTRODUCTION
Angular size is a measure normally used to show how large a celestial body appears from earth.
The angular size, also known as the apparent size, angular diameter, or apparent diameter, is
the amount of space the object occupies in the observer's field of view. The angular diameters
of celestial bodies as seen from earth are typically very small. Therefore, the Small Angle
Formula can be used to calculate the actual size of the object by measuring its angular size and
finding the distance to it [1, 2].
Let us now limit our discussion to the sun and moon. The sun and moon, which appear to be the
biggest heavenly bodies in the earth's sky, have approximately equal apparent sizes of around
0.5° [3]. However, at present, the angular sizes of the sun and moon and their respective
distances are measured using complex and precise instruments. Using the Small Angle Formula,
their actual sizes are hence obtained. Now the question is whether the current apparent and
actual diameters of the sun and moon represent their respective spherical sizes. We must
consider this question in light of the well-known observation that objects appear to taper off
progressively toward the distant end when viewed. But due to the unique geometry of the
sphere, its apparent tapering is not readily noticeable. Nevertheless, the apparent tapering of
the sun and moon is worked out in this study, and their sizes are consequently adjusted.
Why Apparent Tapering of the Sphere is not Readily Noticeable
Our eyes gather information about the size, shape, location, brightness, clarity, and movement
of objects around us, which are then displayed as two-dimensional images on the retina. Our
brain perceives the visuals as three-dimensional with the help of depth perception. The relative
size is one of the indicators for depth perception. In other words, objects of the same size are
perceived as being closer when they are larger and distant when they are smaller [4]. For
example, Figure 1(a & b) [5, 6] illustrates how road tunnels, which are typically of regular width
Page 2 of 10
148
Transactions on Engineering and Computing Sciences (TECS) Vol 13, Issue 1, February - 2025
Services for Science and Education – United Kingdom
and height, appear to taper off gradually in the forward direction. However, due to two unique
aspects of the sphere's geometry, apparent tapering is not easily obvious for spherical objects.
The first aspect is that for the sphere, unlike for regular 3D objects, the relative size indicator
of depth perception is reversed, as illustrated in Figure 2. The nearest to the viewer will be a
point of zero angular size. Then geometrically, the sphere enlarges in the forward direction at
a rate much higher than that of apparent tapering. The second aspect is that the sphere as a
unique 3D object does not have any edges or vertices. It is a round 3D shape with all the points
on its surface at equal distances from the center [7]. How much of the sphere can be seen
depends on the angular diameter [8]. For very small angular diameters, such as that of the sun
or moon, we can see almost a whole hemisphere. So, while the sphere is rotating, as illustrated
in Figure 2, an apparently tapered hemisphere will be continuously generated symmetrically
around the line connecting the view point and the center of the sphere. Therefore, the view will
constantly be the same circular disc, which makes the apparent tapering unnoticeable.
a b
Figure 1: (a) Sharqiyah Tunnel2, Oman [5], (b) Wadi Al Helo Tunnel, UAE [6].
Figure 2: Apparent tapering of a sphere compared to a regular tunnel.
Apparent Tapering of the Sun and Moon and their Adjusted Sizes
First, to form a rough idea about how the sun and moon taper off, let us imagine that a rotating
sphere is circumscribed by a right circular hollow cylinder, as illustrated in Figure 3. Now,
Viewer
Regular
Tunnel
Rotating
Sphere
Object View
Zero Angular Size
of the Closer End
Angular Size of
the Closer End
Angular Size of
the Distant End
Angular Size of
the Distant End
Page 3 of 10
149
Abunaieb, S. (2025). The Sun and Moon are Larger by About 1.4% than we Think. Transactions on Engineering and Computing Sciences, 13(1). 147-
156.
URL: http://dx.doi.org/10.14738/tecs.131.18307
looking down the axis of the cylinder, the angular diameter is assumed to be as small as that of
the sun or moon. In this case, the observer can see almost up to the middle plane [8], where the
cylinder’s internal wall is tangential to the sphere. The cylinder will appear to taper off
gradually toward the distant end in relation to the closer end. Thus, we can imagine that the
rotating sphere contained within the cylinder will likewise eventually taper and apparently
become egg-shaped, as illustrated two-dimensionally by the oval shape in Figure 3. The base
side of the apparently egg-shaped sphere will be seen as a circular disc, as represented by the
yellow circle whose apparent size is equal to the minor diameter of the tapered sphere. The
white ring around it is a view of the apparently tapered front half of the cylinder up to the minor
axis of the tapered sphere. All of the back half, including the middle plane, will be blocked by
the base of the apparently egg-shaped sphere.
Figure 3: Apparent tapering of a sphere circumscribed by a right circular hollow cylinder.
Let's now examine the cylinder's tapering, followed by the combined cylinder and
circumscribed sphere's tapering. In Figure 4, the side view of the cylinder's front half is
Middle
Plane/Axis of
the Sphere
Direction of View
Imaginary Side View of The
Apparently Tapered Sphere
and Cylinder
Bottom View of the
Rotating Apparently
Tapered Sphere
Bottom View the
Apparently
Tapered Front Half
of the Cylinder
Minor Axis of the
Tapered Sphere
Major Axis of the
Tapered Sphere/
Axis of the Cylinder
Page 4 of 10
150
Transactions on Engineering and Computing Sciences (TECS) Vol 13, Issue 1, February - 2025
Services for Science and Education – United Kingdom
geometrically represented by the rectangle ABCD. The radius and depth of the cylinder are
equal to the radius of the circumscribed sphere, R1. The distant end of the front half of the
cylinder is the middle plane, which is blocked, as explained above, by the plane of the minor
axis of the apparently tapered sphere. However, for the very small apparent sizes of the sun or
moon, the difference between the apparent size of the minor axis and that of the middle plane
is not expected to be that significant. Therefore, the angular size of the distant end of the half
cylinder, Θ2, can be approximated to that of the minor axis of the apparently tapered sun or
moon. The rays of the angle Θ2 intersect with the line AB at points E and G, as seen in Figure 4.
Thus, the relative size comparison of the cylinder's far and closer ends equals the linear size of
segment EG to that of line AB. The front view, represented by the two concentric circles on the
right, can then be drawn by projecting the points A, E, F, G, and B horizontally. The two
concentric circles represent how the image lands on the retina. The grey annulus is the
seemingly tapering inner wall, and our brain will interpret the outer circle, with radius R1, as
the closer end and the inner one, with radius R2, as the farther one.
Figure 4: Apparent tapering of a right circular hollow cylinder.
It is obvious that apparent tapering of straight regular objects is of a constant gradient. In the
triangle ADE, as shown in Figure 4, AE is equivalent to ΔR2. which is the difference between R1
and R2. AD is equivalent to the depth R1. Angle ADE equals Θ2/2, since angle DOF, which is half
of Θ2, alternates with angle ADE. The tapering gradient, m, of the aforementioned half cylinder
can, hence, be found using the following formulae:
∆R2 = R1 − R2 (1)
m = ∆R2 /R1 = AE ⁄ AD = tan(θ2/2) (2)
As assumed above, Θ2 is equal to the apparent size of the sun or moon, which is a small angle.
Hence, provided that Θ2 is in radians, using the Small Angle Formula [9], Equation (2) can be
rewritten as follows:
m = tan(θ2 /2) ≈ (θ2/2) (3)
R1
Θ2
Θ2/2
E
B C
A D
F
R1
G H
h
Δ Rh
Δ R2
R2
Rh
O Θ1
K
Θ2/2
Page 5 of 10
151
Abunaieb, S. (2025). The Sun and Moon are Larger by About 1.4% than we Think. Transactions on Engineering and Computing Sciences, 13(1). 147-
156.
URL: http://dx.doi.org/10.14738/tecs.131.18307
∴ m ≈ (θ2/2) (4)
The amount of apparent tapering, ΔRh, at any depth, h, can hence be expressed as follows:
∆Rh = mh ≈ (θ2⁄2)h (5)
Apparent tapering rate, i.e. apparent tapering per unit length of the radius R1 at any depth h, is
hence:
∆Rh/R1 = (θ2⁄2)h/R1 (6)
Let us now consider the apparent tapering of the combined cylinder and circumscribed sphere.
The circle of radius R1 in Figure 5 (a) is a two-dimensional geometric representation of the
sphere. The oval shape represents the seemingly tapering sphere. The rectangle ABCD, which
appears to taper to the trapezoid ABEF, represents the front half of the cylinder. Starting at
point O, the angle φ is obtained by rotating the radius R1 in a counterclockwise direction for one
revolution. The circumscribed sphere represented by the circle in Figure 5 (a) can alternatively
be thought of as a set of successively connected spherical circles that are aligned along the axis
of the cylinder. In any perpendicular plane along the axis of the cylinder, e.g., plane XX at the
random depth h, the radius of the spherical circle is geometrically equal to R1 sin φ. This
spherical circle is contained within the cross section of the right circular cylinder in the same
plane as illustrated in Figure 5 (b). Obviously, the tapering per unit length of the cylinder's
radius Δ Rh/R1, as given above by Equation (6), will apply to the spherical circle contained in
the same plane
Figure 5: (a) The combined cylinder and circumscribed sphere's tapering. (b) Plane XX.
φ
x
Direction of View
h R1 cos
φ
R1 sin φ
A O
F
B
D E C
Middle
Plane/Axis of the
Sphere
Minor Axis of the
Tapered Sphere
X X
x
Plane XX
Δ x
Page 6 of 10
152
Transactions on Engineering and Computing Sciences (TECS) Vol 13, Issue 1, February - 2025
Services for Science and Education – United Kingdom
With reference to Figure 5 (a & b), it is possible to express the variables h, ΔRh/R1, Δx, and
consequently x in terms of the independent variable angle φ, the geometrically adjusted radius
of the sphere R1, and its apparent size Θ2.
As illustrated in Figure 5 (a), the depth, h, is given by:
h = R1 − R1 cos φ = R1(1 − cos φ) (7)
From Equations (6) and (7), the apparent tapering of the cylinder per unit length of the
geometric radius R1 is:
∆Rh/R1 = (θ2⁄2)R1(1 − cos φ)/R1 = (θ2⁄2)(1 − cos φ) (8)
In any given plane, the difference, Δx, between the geometric radius of the spherical circle and
its tapered radius can be obtained by multiplying the tapering per unit length given in Equation
(8) by the spherical circle’s radius, R1 sin φ, as follows:
∆ x = (θ2⁄2)(1 − cos φ)R1 sin φ (9)
From Figure 5 (b):
x = R1 sin φ − ∆ x (10)
From (9) and (10):
x = R1 sin φ − (θ2⁄2)(1 − cos φ)R1 sin φ = R1 sin φ [1 − (θ2⁄2) + (θ2⁄2) cos φ] (11)
From (11):
x = (1 − θ2⁄2)R1 sin φ + (θ2⁄2)R1 sin φ cos φ (12)
Equation (12) shows that the spherical circle’s radius x is a function of the variable angle φ, the
geometric radius of the sphere, R1, and the apparent size of the sun or moon, Θ2. As seen in
Figure 5 (a), the spherical circle's radius x increases from zero at φ equal to zero to a maximum
equal to the minor radius of the oval shape, Ro, at a value of φ close to 90°. The minor radius Ro
represents the currently recorded radius of the sun or moon. The plane of the minor radius Ro
is a little bit below the center of the sphere. This plane, as explained earlier and as illustrated in
Figures 3 and 5 (a), blocks all of the top part, including the central geometric plane of the
combined sphere and cylinder. The angle φ at which x is maximum, i.e., at which x is equal to
Ro, can be found by equating the first-order derivative of Equation (12) to zero [10].
Using differentiation of trigonometric functions [11], the derivative of Equation (12) is:
dx dφ = (1 − θ2⁄2)R1 cos φ + (θ2⁄2)R1(cos2 ⁄ φ − sin2φ) (13)
Page 7 of 10
153
Abunaieb, S. (2025). The Sun and Moon are Larger by About 1.4% than we Think. Transactions on Engineering and Computing Sciences, 13(1). 147-
156.
URL: http://dx.doi.org/10.14738/tecs.131.18307
From basic trigonometric identities, sin2φ + cos2φ = 1 for any angle φ. Therefore, sin2φ in
Equation (13) can be replaced with (cos2φ – 1). Equating dx /dφ to zero, we then obtain:
(1 − θ2⁄2)R1 cos φ + (θ2⁄2)R1
(2cos2φ − 1) = 0 (14)
Equation (12) can be rewritten as:
θ2cos2φ + (1 − θ2⁄2) cos φ − θ2⁄2 = 0 (15)
The value of the unknown x in a quadratic equation of the form ax2 + bx + c = 0 may be found
using the Quadratic Formula [12], x = (-b ± √ (b2 - 4ac))/(2a), where the coefficients a, b, and c
are real or complex numbers. Equation (15) above is a quadratic equation with the coefficients
a = Ɵ2, b = (1 - Ɵ2/2), and c = (- Ɵ2/2), where Ɵ2 is in radians and cos φ is the unknown x.
Therefore, cos φ can be obtained as follows:
cos φ = {−(1 − θ2⁄2) ± √(1 − θ2⁄2)
2 − 4θ2
(− θ2⁄2) }⁄2θ2 (16)
Equation (16) can be rewritten as below:
cos φ = {−(1 − θ2⁄2) ± √(1 − θ2⁄2)
2 + 2(θ2)
2 }⁄2θ2 (17)
Hence,
φ = cos−1
〈{−(1 − θ2⁄2) ± √(1 − θ2⁄2)
2 + 2(θ2)
2 }⁄2θ2
〉 (18)
RESULTS AND DISCUSSION
In the above equation, Ɵ2, as was previously supposed, is alternately the apparent diameter of
the sun and moon. From the Sun Fact Sheet [13], the apparent diameter of the sun, Ɵ2, is 1919
arc seconds. For the moon, the apparent diameter is 1896 arc seconds [14]. Converting the arc
seconds into radians [15], the values of Ɵ2 for the sun and moon are 0.009304 and 0.009192
radians, respectively. In the domain 0 ≤ φ ≤ 90°, the angle φ at which x is maximum, i.e., x = Ro,
can hence be obtained for the sun and moon by substituting their above respective apparent
diameters in Equation (18). Equation (18) as a cosine function has a range between -1 and 1
[16]. Considering the values within the range, the values of the angle φ at which x is maximum
(x = Ro) for the sun and moon are 89.735 and 89.732°, respectively.
Now, substituting the above respective values of the angle Ɵ2 in radians and the angle φ in
degrees in Equation (12), we obtain a ratio of the currently recorded radius, Ro, to the
geometrically corrected radius, R1, of the sun and moon, respectively, as follows:
For the sun:
R0/R1 =0.99536 (19)
Page 8 of 10
154
Transactions on Engineering and Computing Sciences (TECS) Vol 13, Issue 1, February - 2025
Services for Science and Education – United Kingdom
From (19):
R1/Ro =1.00466 (20)
From (20), the ratio of the geometrically adjusted volume V1 to the currently recoded volume
Vo of the sun is Hence:
V1/Vo = (1.00466) ^3=1.014 (21)
Hence:
(V1 − Vo) /Vo% = (1.014 − 1) ∗ 100/1 = 1.41% (22)
Now, as evident from Equation (22), the sun's geometrically adjusted volume is larger than the
currently recorded volume by 1.4%. Therefore, the sun's mass, based on its mean density, is
also 1.4% larger. With reference to the Sun Fact Sheet [14], the sun/earth mass ratio is thus
increased from 333,000 to 337,662. In other words, the geometrically adjusted mass of the sun
is bigger by 4,662 times the mass of Earth.
For the moon, and as the values of the angles Ɵ2 and φ are almost equal to those of the sun, the
following ratios of the radii and volumes are almost equal to those of the sun:
R0/R1 =0.99541 (23)
From (23):
R1/Ro =1.00461 (24)
From (24)
V1/Vo = (1.00461) ^3=1.014 (25)
From (25), the ratio of the geometrically adjusted volume V1 to the currently recoded volume
Vo of the moon is Hence:
(V1 − Vo) /Vo% = (1.014 − 1) ∗ 100/1 = 1.39% (26)
Similar to the sun, the moon's geometrically adjusted volume is approximately 1.4% greater
than its currently recorded volume. Therefore, the moon's mass, based on its mean density, is
also 1.4% larger. But according to the Moon Fact Sheet [15], the moon/earth ratio is just 0.0123.
As a result, the adjusted mass is bigger by only 0.000172 times that of Earth.
CONCLUSION
The fundamental idea of this work was the apparent tapering of objects toward their far end.
As explained in Section 2 of this paper, the apparent tapering of spherical objects is difficult to
Page 9 of 10
155
Abunaieb, S. (2025). The Sun and Moon are Larger by About 1.4% than we Think. Transactions on Engineering and Computing Sciences, 13(1). 147-
156.
URL: http://dx.doi.org/10.14738/tecs.131.18307
see, but it can be realized when a sphere is geometrically encased in a right circular cylinder or
even a tunnel. Following the apparent tapering of the cylinder or tunnel toward its far end, one
can imagine the eventual tapering of the encased sphere. This idea was used in Sections 3 and
4 to calculate the apparent tapering of the sphere encircled by a right circular cylinder. It was
easy to calculate the tapering rate of any plane of the cylinder along its axis because the tapering
of regular, straight, hollow objects is obviously a linear function of the plane's depth with
respect to the closer end. Thus, the tapering amount is the product of the depth and the tapering
rate. It goes without saying that the radius of the cylinder will taper uniformly over the plane's
radius. The sphere's confined portion will thereafter taper at the same pace. Then, with the aid
of basic geometry, it has been demonstrated that the volume of the sun and moon is 1.4%
greater than what is commonly understood. Due to their nearly equal apparent diameters, the
sun and moon have shown almost equal corrected to recorded volume ratios.
References
[1]. University of Iowa (n.d.) Imaging the Universe. Available at:
https://itu.physics.uiowa.edu/labs/foundational/angular-size (Accessed: 9 October 2024).
[2]. Perfect Astronomy (n.d.) What is Angular Size in Astronomy? Available at:
https://perfectastronomy.com/astronomy-course/angular-size/ (Accessed: 30 October 2024).
[3]. Jeffery, D. (n.d.) The Great Coincidence. Available at:
https://www.physics.unlv.edu/~jeffery/astro/moon/sun_moon_angular.html (Accessed: 7 October
2024).
[4]. All About Vision (n.d.) Depth Perception: How Do We See in 3D? Available at:
https://www.allaboutvision.com/eye-care/eye-anatomy/depth-perception/ (Accessed: 19 October 2024).
[5]. Oman Observer (n.d.) $130m Funding Support for Oman Tunnels Project. Available at:
https://www.omanobserver.om/article/12681/Main/130m-funding-support-for-oman-tunnels-project
(Accessed: 18 October 2024).
[6]. Foursquare (n.d.) Wadi Al Helo Tunnel. Available at: https://foursquare.com/v/wadi-al-helo- tunnel/51b1a55a498eab79364e28c3 (Accessed: 18 October 2024).
[7]. Byju’s (n.d.) Sphere. Available at: https://byjus.com/maths/sphere/ (Accessed: 23 October 2023).
[8]. Wikipedia (n.d.) Angular Diameter. Available at: https://en.wikipedia.org/wiki/Angular_diameter
(Accessed: 25 October 2024).
[9]. University of Iowa (n.d.) Small Angle Formula. Available at: https://itu.physics.uiowa.edu/glossary/small- angle-formula (Accessed: 24 December 2024).
[10]. Byju’s (n.d.) Maxima and Minima in Calculus. Available at: https://byjus.com/jee/maxima-and-minima-in- calculus/ (Accessed: 7 December 2024).
[11]. Cuemath (n.d.) Differentiation of Trigonometric Functions. Available at:
https://www.cuemath.com/trigonometry/differentiation-of-trigonometric-functions/ (Accessed: 24
December 2024).
[12]. Wikipedia (n.d.) Quadratic Formula. Available at: https://en.wikipedia.org/wiki/Quadratic_formula
(Accessed: 26 December 2024).
Page 10 of 10
156
Transactions on Engineering and Computing Sciences (TECS) Vol 13, Issue 1, February - 2025
Services for Science and Education – United Kingdom
[13]. NASA (n.d.) Sun Fact Sheet. Available at: https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
(Accessed: 5 October 2024).
[14]. NASA (n.d.) Moon Fact Sheet. Available at: https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html
(Accessed: 28 December 2024).
[15]. Inch Calculator (n.d.) Seconds of Arc to Radians Converter. Available at:
https://www.inchcalculator.com/convert/arcsecond-to-radian/ (Accessed: 8 February 2025).
[16]. GeeksforGeeks (n.d.) Domain and Range of Trigonometric Functions. Available at:
https://www.geeksforgeeks.org/domain-and-range-of-trigonometric-functions/ (Accessed: 8 February
2025).