Square on a Sphere

We'll divide the square into two equal triangles and calculate their area. Then, the angles of the triangles are $\alpha=\beta=50^{\circ},\, \gamma=100$

Note that the area of a spherical triangle depends only on the sum of the interior angles and the radius of the sphere, by the formula:

$area(triangle)=R^2(\alpha+\beta+\gamma-\pi$ )

Hence, $area(square)=2*100^2*((20/180)*\pi)\approx 6981.317\Rightarrow \fbox{6981.32}$

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Now, where does the formula come from?

We have to imagine how the sides of a triangle divide the sphere. Let ABC be a spherical triangle. The sides of the triangle determine three great circumferences on the sphere. Then, the triangle ABC and the opposite triangle across the segment BC form a lune, $L_a$ , ABC and the opposite triangle across AC form another lune $L_b$ and ABC and the opposite triangle across AB form another lune $L_c$ . Notice that each lune has an opposite lune equal in area (because two great circumferences determine two equal lunes) and those cover the rest of the sphere, so the area covered by the three lunes is half the area of the sphere. Hence,

$2\pi R^2=(area(L_a)-area(ABC))+(area(L_b)-area(ABC))+(area(L_c)-area(ABC))+area(ABC)$ $2\pi R^2=area(L_a)+area(L_b)+area(L_c)-2area(ABC)$

Also, the area of each lune is proportional to its angle, so

$2\pi R^2=2\alpha R^2+2\beta R^2+2\gamma R^2-2area(ABC)$

$area(ABC)={ R^2(\alpha+\beta+\gamma-\pi)}$