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

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

$=268\;m^{3}$

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