Spherical Wedge and Spherical Lune Formula

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 ?


R = 8 m
θ = π/4 radians

Using the formula:

\(\begin{array}{l}Volume=\frac{2}{3}\:R^{3}\theta\end{array} \)

\(\begin{array}{l}Volume=\frac{2}{3}\times 8^{3}\times \frac{\pi}{4}\end{array} \)

\(\begin{array}{l}=268\;m^{3}\end{array} \)


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