# Spherical Trigonometry

**Spherical Triangle**

Any section made by a cutting plane that passes through a sphere is circle. A great circle is formed when the cutting plane passes through the center of the sphere. Spherical triangle is a triangle bounded by arc of great circles of a sphere.

Note that for spherical triangles, sides *a*, *b*, and *c* are usually in angular units. And like plane triangles, angles *A*, *B*, and *C* are also in angular units.

**Sum of interior angles of spherical triangle**

The sum of the interior angles of a spherical triangle is greater than 180° and less than 540°.

**Area of spherical triangle**

The area of a spherical triangle on the surface of the sphere of radius *R* is given by the formula

Where *E* is the spherical excess in degrees.

**Spherical excess**

or

$\tan \frac{1}{4}E = \sqrt{\tan \frac{1}{2}s~\tan \frac{1}{2}(s - a)~\tan \frac{1}{2}(s - b)~\tan \frac{1}{2}(s - c)}$

Where $s = \frac{1}{2}(a + b + c)$

**Spherical defect**

Note:

In spherical trigonometry, earth is assumed to be a perfect sphere. One minute (0° 1') of arc from the center of the earth has a distance equivalent to one (1) nautical mile (6080 feet) on the arc of great circle on the surface of the earth.

1 nautical mile = 6080 feet

1 statute mile = 5280 feet

1 knot = 1 nautical mile per hour