Chapter 4: Geometry

4.18.1.3 Spherical segment and lune
Let the radius be Ö (Figure 4.40, right). The area of the curved region (called a
spherical segment or lune)is¾Ö ¾ , the angle being measured in radians. The volume
of the segment is ¾ ¿ Ö¿ .
4.18.1.4 Volume and area of spheres
If the volume of an Ò-dimensional sphere of radius Ö is Î Ò´Öµ and its surface area is
Ë Ò´Öµ, then
Î Ò´Öµ ¾Ö¾
Ò
ÎÒ ¾´Öµ ¾Ò¾Ö ¡ Ò
Ò
Ò¾ Ö
¡ Ò
Ò
Ò
¾
¾
Ë Ò Ò´Öµ
Ö Î Ò´Öµ
Ö Î Ò´Öµ℄
(4.18.11)
Hence, the area of a circle is Î ¾ Ö ¾ ¿½½Ö ¾ , the volume of a 3-dimensional
sphere is Î ¿ ¿ Ö¿ ½Ö ¿ , the volume of a 4-dimensional sphere is Î
½
¾ ¾ Ö ¿Ö , the circumference of a circle is Ë ¾ ¾Ö, and the surface area
of a sphere is Ë ¿ Ö ¾ .
For large values of Ò,
Î Ò´Öµ Ò ´Ò·½µ¾
Ô
´¾µ Ò¾ Ö Ò (4.18.12)
4.19 SPHERICAL GEOMETRY & TRIGONOMETRY
The angles in a spherical triangle do not have to add up to 180 degrees. It is possible
for a spherical triangle to have 3 right angles.
4.19.1 RIGHT SPHERICAL TRIANGLES
Let , , and be the sides of a right spherical triangle with opposite angles , ,
and , respectively, where each side is measured by the angle subtended at the center
of the sphere. Assume that ¾ ¼ Æ (see Figure 4.41, left). Then,
×Ò ØÒ ÓØ ×Ò ×Ò
×Ò ØÒ ÓØ ×Ò ×Ò
Ó× Ó× ÓØ Ó× Ó×
Ó× ØÒ ÓØ Ó× ×Ò
Ó× ØÒ ÓØ Ó× ×Ò
4.19.1.1 Napier’s rules of circular parts
Arrange the five quantities , , co- (this is the complement of ), co-, co-
of a right spherical triangle with right angle at , in cyclic order as pictured in
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