Spherical Wedge and Spherical Lune

A spherical wedge is a solid formed by revolving a semi-circle about its diameter by less than 360°. Spherical Lune is the curve surface of the wedge, it is a surface formed by revolving a semi-circular arc about its diameter by less than 360°.
 

Spherical Wedge and Spherical Lune

 

Formula for Spherical Wedge and Lune
The formula for spherical wedge and Lune can be found by ratio and proportion, meaning, spherical wedge is similar to the sphere and spherical Lune is similar to spherical surface.
 

Volume of wedge, Vwedge
Volume of wedge / central angle = Volume of sphere / 1 revolution

$\dfrac{V_{wedge}}{\theta_{deg}} = \dfrac{\frac{4}{3}\pi R^3}{360^\circ}$

$V_{wedge} = \frac{1}{270} \pi R^3 \theta_{deg}$

$V_{wedge} = \frac{2}{3} R^3 \theta_{rad}$

 

Area of Lune, Alune
Area of Lune / central angle = Area of sphere / 1 revolution

$\dfrac{A_{lune}}{\theta_{deg}} = \dfrac{4\pi R^2}{360^\circ}$

$A_{lune} = \frac{1}{90} \pi R^2 \theta_{deg}$

$A_{lune} = 2 R^2 \theta_{rad}$

 

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