Area of Spherical Lune

Theorem

Let $S$ be a sphere of radius $r$.

Let $\CC_1$ and $\CC_2$ be two great circles on $S$ such that the spherical angle between $\CC_1$ and $\CC_2$ is $\theta$.

Let $\LL$ be the lune described by $\CC_1$ and $\CC_2$.

Then the area of $\LL$ is given by:

$\map \Area \LL = 2 r^2 \theta$

or, if $\theta$ is measured in degrees:

$\map \Area \LL = \dfrac {4 \pi r^2 \theta} {360}$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): lune: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): lune: 1.