• Sides as Angles: Unlike plane triangles, the sides of a spherical triangle ($a, b, c$) are measured as central angles subtended at the center of the sphere, typically in degrees or radians. The actual linear length of a side is the arc length $s = R \times (\text{side in radians})$, where $R$ is the sphere's radius.
• Vertices: The angles of the triangle ($A, B, C$) are the dihedral angles between the planes of the great circles intersecting at the vertices.
• Sum of Angles: The sum of the angles $A + B + C$ is strictly greater than $180^\circ$ and less than $540^\circ$.
• Spherical Excess: The amount by which the sum of the angles exceeds $180^\circ$ is called the spherical excess ($E$).

The area of a spherical triangle is directly proportional to its spherical excess:

Every spherical triangle $ABC$ has a corresponding polar triangle $A'B'C'$, formed by the poles of the great circles that make up the sides of the original triangle.

• The angles of one triangle are supplementary to the corresponding sides of its polar triangle.
• $A' = 180^\circ - a$, $B' = 180^\circ - b$, $C' = 180^\circ - c$
• $a' = 180^\circ - A$, $b' = 180^\circ - B$, $c' = 180^\circ - C$

This property is extremely useful because it allows us to convert a problem where we know three angles (AAA) into a problem where we know three sides (SSS) of the polar triangle, which can then be solved using the Law of Cosines.

Spherical trigonometry was primarily developed by Islamic astronomers and mathematicians (like Al-Jayyani and Nasir al-Din al-Tusi) during the Islamic Golden Age to solve problems related to astronomy, timekeeping, and finding the Qibla (the direction of Mecca). These early mathematical foundations later became essential for global maritime exploration and modern celestial navigation.

If angle $C = 90^\circ$, the remaining five parts are arranged in a specific order in a circle divided into five sectors:

• The two sides forming the right angle: $a$ and $b$.
• The complements of the remaining parts: $90^\circ - A$ (or $co-A$), $90^\circ - c$ (or $co-c$), and $90^\circ - B$ (or $co-B$).

The order around the circle is: $a$, $b$, $co-A$, $co-c$, $co-B$.

Napier's Rules

Select any one of the five parts as the middle part. The two adjacent parts are adjacent parts, and the remaining two are opposite parts.

• Rule 1: The sine of the middle part equals the product of the tangents of the adjacent parts. (Sin-Tan-Ad)

• Rule 2: The sine of the middle part equals the product of the cosines of the opposite parts. (Sin-Cos-Op)

Spherical Law of Sines

Spherical Law of Cosines for Sides

Spherical Law of Cosines for Angles

Napier's Analogies are useful formulas for solving oblique spherical triangles when you are given two sides and the included angle (SAS) or two angles and the included side (ASA). They express half-angles and half-sides.

Often attributed to Gauss but published earlier by Delambre, these equations relate the functions of half the sides to functions of half the angles. They are frequently used alongside Napier's Analogies to check spherical calculations.

Unlike plane triangles where the sum of interior angles is always exactly $180^\circ$ (or $\pi$ radians), the sum of the angles of a spherical triangle always exceeds $180^\circ$. This difference is called the Spherical Excess ($E$).

(where angles are strictly in radians)

Girard's Theorem states that the surface area of a spherical triangle is directly proportional to its spherical excess:

This principle is fundamental in calculating land areas over large sections of the Earth.

• Latitude ($\phi$): The angle north or south of the equator. The polar distance (colatitude) is $90^\circ - \phi$.
• Longitude ($\lambda$): The angle east or west of the Prime Meridian.

To find the great circle distance between two points $P_1(\phi_1, \lambda_1)$ and $P_2(\phi_2, \lambda_2)$, we form a spherical triangle with the North Pole ($N$).

• Side $P_1N = 90^\circ - \phi_1$
• Side $P_2N = 90^\circ - \phi_2$
• Angle $N = |\lambda_1 - \lambda_2|$ (Difference in longitude)

Applying the Spherical Law of Cosines for the side $P_1P_2$ ($d$ in angular measure):

The actual distance is $D = R \times d$ (where $d$ is in radians).

While the Spherical Law of Cosines works mathematically, it can suffer from severe rounding errors on computers when calculating small distances (where $\cos d \approx 1$). The Haversine formula is numerically stable for all distances and is the standard method used in surveying, GPS, and geodesy.

Where the haversine function is defined as $\text{hav}(\theta) = \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{2}$.

Solving for the distance $D$ (where $D = R \cdot d$, and $d$ is in radians):

While spherical trigonometry assumes the Earth is a perfect sphere, it is actually an oblate spheroid (slightly squashed at the poles).

For short distances or basic celestial navigation, assuming a mean radius of $R \approx 6371 \text{ km}$ is sufficient. However, for precision modern geodesy, surveying, and the Global Positioning System (GPS), civil engineers must use the WGS 84 (World Geodetic System 1984) reference ellipsoid.

On an ellipsoid, the shortest path between two points is called a geodesic, which is much more complex to calculate than a great circle. When extreme accuracy is required, equations like Vincenty's formulae are used instead of the Haversine formula to account for the Earth's oblateness.