Spherical Wedge and Spherical Lune Formula

A solid formed by revolving a semicircle about its diameter with less than 360 degrees, then it is called a spherical wedge. Spherical lune is the area on a sphere surrounded by two half circles that meet at antipodal points.When we say antipodal, we mean it is the point on the surface of a sphere which is diametrically opposite to it. If we draw a line from one point to another then it will form a straight line. The formula for spherical wedge is to calculate volume, arc length and surface area.

The formulas are:

\[\large Volume=\frac{2}{3}\:R^{3}\theta\]

\[\large Surface\;Area=2R^{2}\theta\]

\[\large Arc\;Length\;at\;the\;equator=R\theta\]

Solved Example

Question: Find the volume of the spherical wedge with radius 8 m and the dihedral angle of the wedge 45 degrees ?

Solution:

Given,
R = 8 m
θ = 45 degrees

Using the formula: $Volume=\frac{2}{3}\:R^{3}\theta$

$Volume=\frac{2}{3}\times 8^{3}\times 45$

$=268\;m^{3}$


Practise This Question

Which one of the following is/are true in the formation of emulsion?

i) the liquid in larger proportion act as dispersion of medium

ii) the liquid with higher surface tension forms the dispersion medium

iii) the liquid in which the emulsifying agent is more soluble forms the dispersion medium

iv) an emulsion is formed by vigorous shaking of two immiscible liquids