$\angle CAB=$
$\angle CDB=$

Given that the area of a unit sphere (r=1) is given by the formula

we have that Area(Lune ACDB)

$A=4\pi r^2=4\pi(1)=4\pi$,

$\approx$

If △ABC is determined by the minor arcs 

$\stackrel\frown ^{AB}$, $\stackrel\frown ^{AC}$,  $\stackrel\frown ^{CB}$,

where        is a segment through the center of the disk, then

$\stackrel\frown ^{CB}$ 

we have that Area(△ABC)           .

$\approx$

A

D

Area(△ ABC) 

c = Circle((0, 0), (1, 0))
O = Center(c)
B_1 = Point(c)
D_1 = Antipode(B_1, c)
f = Segment(B_1, D_1)
C = Point(f)
A_1 = Point(f)
C_1 = Antipode(C, c)
g = Segment(C, C_1)
F = Midpoint(g)
d = Circle(F, C)
A = Intersect(c, d, 2)
D = Intersect(c, d, 1)
h = Segment(A, D)
e = CircumcircleArc(A, C, D)
B = Point(f)
k = CircumcircleArc(A, B, D)
α = EllipticAngleMeasure(B, D, C, c)
β = EllipticAngleMeasure(C, A, B, c)
a: ((β / (360°) * 4) * π)
b: a / 2
text6 = LaTeX(a, true, false)
text10 = LaTeX(b, true, false)
list1: {e, k}
K = Point(list1)
M: K
loc1 = Locus(M, K)
i: ((Min[α, (2 * π) - α] * 180 / π))°
text2 = LaTeX(i, true, false)
c_1 = Circle((0, 0), (1, 0))
O_1 = Center(c_1)
B_2 = Point(c_1)
D_2 = Antipode(B_2, c_1)
f_1 = Segment(B_2, D_2)
C_2 = Point(f_1)
A_2 = Point(f_1)
C_3 = Antipode(C_2, c_1)
g_1 = Segment(C_2, C_3)
F_1 = Midpoint(g_1)
d_1 = Circle(F_1, C_2)
D_3 = Intersect(c_1, d_1, 1)
B_3 = Point(f_1)
γ: ((Min[β, (2 * π) - β] * 180 / π))°
text3 = LaTeX(γ, true, false)
text11 = LaTeX(i, true, false)
text12 = LaTeX(γ, true, false)

Tool: Antipode
Input: C, c
Output: E

A = Center(c)
f = Line(A, C)
g = OrthogonalLine(A, f)
B, D = Intersect(c, g)
h = Line(C, D)
i = OrthogonalLine(D, h)
E = Intersect(f, i)

Tool: Elliptic Line (command EllipticLine)
Input: B, C, c
Output: d

D = Antipode(C, c)
d = Circle(B, C, D)

Tool: Elliptic Semiline (command EllipticSemiline)
Input: B, C, c
Output: e

d = EllipticLine(B, C, c)
D = Intersect(c, d, 2)
E = Intersect(c, d, 1)
e = CircumcircleArc(D, B, E)

Tool: Elliptic Segment (command EllipticSegment)
Input: B, C, c
Output: h

d = EllipticSemiline(B, C, c)
D = Center(d)
e = Circle(D, C)
f = Segment(C, B)
E = Midpoint(f)
g = Ray(D, E)
F = Intersect(e, g, 1)
h = CircumcircleArc(C, F, B)

Tool: Elliptic Angle Measure (command EllipticAngleMeasure)
Input: A, B, C, c
Output: β

E = Center(c)
f = Line(B, E)
g = Segment(A, B)
j = OrthogonalLine(E, f)
M = Midpoint(g)
D, X = Intersect(c, j)
k = OrthogonalLine(M, g)
h = Line(B, X)
I = OrthogonalLine(X, h)
B' = Intersect(I, f)
i = Segment(B, B')
C_1 = Midpoint(i)
m = OrthogonalLine(C_1, i)
D_1 = Intersect(k, m)
l = Segment(D_1, B)
n = OrthogonalLine(B, l)
o = OrthogonalLine(A, n)
E_1 = Intersect(o, n)
p_1 = Segment(B, C)
F = Midpoint(p_1)
p = OrthogonalLine(F, p_1)
G = Intersect(p, m)
q_1 = Segment(G, B)
q = OrthogonalLine(B, q_1)
r = OrthogonalLine(C, q)
H = Intersect(r, q)
β = Angle(H, B, E_1)