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Is there any analytic or generalized formula to calculate area covered by any rectangle having length $l$ & width $b$ each as a great circle arc on a spherical surface with a radius $R$? i.e. How to find the area $A$ of rectangle in terms of length $l$, width $b$ and radius $R$ ($A=f(l, b,R)$)?

Note: Spherical rectangle is a quadrilateral having equal opposite sides but non-parallel & all the interior angles are equal in magnitude & each one is greater than $90^\circ$.

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    $\begingroup$ What do you mean by "rectangle"? Do you just want a quadrilateral, or some stronger condition? (Note that you cannot have four right angles in a quadrilateral on the sphere.) $\endgroup$ Mar 25, 2015 at 13:16
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    $\begingroup$ Yes, a quadrilateral having equal opposite sides (but not parallel) each as a great circle arc on a sphere. $\endgroup$ Mar 25, 2015 at 13:21
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    $\begingroup$ Yes, length & width are simply the sides of the rectangle as great circles arcs on the spherical surface. $\endgroup$ May 28, 2015 at 23:56
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    $\begingroup$ To the down-voters and close-voters, notice that the OP has posted an answer. Furthermore, this question is from a time when the context requirement was just being discussed. $\endgroup$
    – robjohn
    May 11, 2020 at 2:58
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    $\begingroup$ Those who have doubts or need any clarification they can ask for details before down voting or voting to close this pretty old question. $\endgroup$ May 11, 2020 at 3:08

7 Answers 7

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Assume we are working on a sphere of radius $1$, or consider the lengths in radians and the areas in steradians.

Extend the sides of length $l$ until they meet. This results in a triangle with sides $$ w,\quad\frac\pi2-\frac l2,\quad\frac\pi2-\frac l2 $$

enter image description here

The Spherical Law of Cosines says that $$ \begin{align} \cos(A) &=\frac{\cos\left(\frac\pi2-\frac l2\right)-\cos\left(\frac\pi2-\frac l2\right)\cos(w)}{\sin\left(\frac\pi2-\frac l2\right)\sin(w)}\\ &=\frac{\sin\left(\frac l2\right)}{\cos\left(\frac l2\right)}\frac{1-\cos(w)}{\sin(w)}\\[6pt] &=\tan\left(\frac l2\right)\tan\left(\vphantom{\frac l2}\frac w2\right) \end{align} $$ One quarter of the spherical excess of the rectangle is $D-\frac\pi2$ and $$ \sin\left(D-\frac\pi2\right)=\tan\left(\frac l2\right)\tan\left(\vphantom{\frac l2}\frac w2\right) $$ Therefore, the area of the rectangle is $$ \bbox[5px,border:2px solid #C0A000]{4\sin^{-1}\left(\tan\left(\frac l2\right)\tan\left(\vphantom{\frac l2}\frac w2\right)\right)} $$ Note that for small $l$ and $w$, this is approximately $lw$; and if $l+w=\pi$ (that is, the rectangle is a great circle), we get an area of $2\pi$ (one half the sphere).

Scaling for a sphere of radius $R$ gives $$ \bbox[5px,border:2px solid #C0A000]{4R^2\sin^{-1}\left(\tan\left(\frac l{2R}\right)\tan\left(\vphantom{\frac l2}\frac w{2R}\right)\right)} $$


Note the similarity to the formula for the area of a spherical right triangle with legs $a$ and $b$: $$ 2\tan^{-1}\left(\tan\left(\vphantom{\frac b2}\frac a2\right)\tan\left(\frac b2\right)\right) $$ or for a sphere of radius $R$, $$ 2R^2\tan^{-1}\left(\tan\left(\vphantom{\frac b2}\frac a{2R}\right)\tan\left(\frac b{2R}\right)\right) $$

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    $\begingroup$ how did you render that graph? $\endgroup$ Dec 20, 2015 at 2:22
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    $\begingroup$ I notice that this answer was unaccepted. Is there something wrong, or something more you need? $\endgroup$
    – robjohn
    May 2, 2020 at 0:20
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    $\begingroup$ No I need to revisit this carefully $\endgroup$ May 2, 2020 at 0:26
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    $\begingroup$ ok id like to accept this but the accept answer button is not visible $\endgroup$ May 10, 2020 at 17:39
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    $\begingroup$ apologies, im clearly out of my mind here... i didn't even ask this question $\endgroup$ May 10, 2020 at 17:50
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On a sphere of radius $R > 0$, a geodesic triangle with interior angles $\theta_{1}$, $\theta_{2}$, and $\theta_{3}$ has area $R^{2}(\theta_{1} + \theta_{2} + \theta_{3} - \pi)$. One way you might proceed, therefore, it to triangulate your geodesic polygon (whatever its actual shape) and sum the areas of its triangular pieces. For a quadrilateral with interior angles $\theta_{1}$, $\theta_{2}$, $\theta_{3}$, and $\theta_{4}$, the area is $$ R^{2}(\theta_{1} + \theta_{2} + \theta_{3} + \theta_{4} - 2\pi). $$


Added (in light of OP's clarifications): If $\theta \leq \pi$ is the interior angle at each vertex of the quadrilateral, then $$ \theta = \pi - \cos^{-1}\left\lvert\tan\frac{\ell}{2R} \tan\frac{b}{2R}\right\rvert, $$ so $$ \text{Area} = R^{2}(4\theta - 2\pi) = R^{2} \left[2\pi - 4\cos^{-1}\left\lvert\tan\frac{\ell}{2R} \tan\frac{b}{2R}\right\rvert\right]. $$ (Particularly, the absolute value of the product of tangents inside the arccos does not exceed unity.)

To see this, it's convenient to work in Cartesian coordinates with the sphere centered at the origin. Denote the vertices of the quadrilateral by $$ v_{1} = (A, B, C),\qquad v_{2} = (A, -B, C),\qquad v_{3} = (A, -B, -C),\qquad v_{4} = (A, B, -C). $$ Of course, $A^{2} + B^{2} + C^{2} = R^{2}$.

A spherical rectangle with vertices in Cartesian coordinates

The plane through the origin, $v_{1}$, and $v_{2}$ has equation $Cx - Az = 0$, and so has unit normal vector $$ n_{1} = \frac{(C, 0, -A)}{\sqrt{A^{2} + C^{2}}} = \frac{(C, 0, -A)}{\sqrt{R^{2} - B^{2}}}. $$ The plane through the origin, $v_{1}$, and $v_{4}$ has equation $Bx - Ay = 0$, and so has unit normal vector $$ n_{2} = \frac{(B, -A, 0)}{\sqrt{A^{2} + B^{2}}} = \frac{(B, -A, 0)}{\sqrt{R^{2} - C^{2}}}. $$

The "large" angle between these planes is the "large" angle between the great circles they determine (because each normal vector $n_{i}$ is tangent to the sphere at $v_{1}$), i.e., the interior angle $\theta$ of the quadrilateral. Taking the ordinary dot product of the unit normals, $$ \cos\theta = n_{1} \cdot n_{2} = \frac{B}{\sqrt{R^{2} - B^{2}}}\, \frac{C}{\sqrt{R^{2} - C^{2}}}. \tag{1} $$

Let $2\psi_{1}$ denote the angle subtended at the center of the sphere by the side from $v_{1}$ to $v_{2}$, and let $\ell_{1} = 2R\psi_{1}$ denote the corresponding side length, so that $\psi_{1} = \ell_{1}/(2R)$. Thinking of $v_{1}$ and $v_{2}$ as vectors in space, the ordinary dot product gives $$ \frac{v_{1} \cdot v_{2}}{R^{2}} = \cos(2\psi_{1}) = 1 - 2\sin^{2} \psi_{1}. $$ On the other hand, using the Cartesian components of these vectors, we have $$ \frac{v_{1} \cdot v_{2}}{R^{2}} = \frac{A^{2} - B^{2} + C^{2}}{A^{2} + B^{2} + C^{2}} = 1 - 2\frac{B^{2}}{R^{2}}. $$ Equating, it follows at once that $B/R = \pm\sin\psi_{1}$. (As a consistency check, the isoceles triangle with the origin, $v_{1}$, and $v_{2}$ as vertices has apex angle $2\psi_{1}$ and base $2B$.)

Similarly, letting $2\psi_{2}$ denote the angle subtended at the center of the sphere by the side from $v_{1}$ to $v_{4}$, and letting $\ell_{2} = 2R\psi_{2}$ denote the corresponding side length, we find $C/R = \pm\sin\psi_{2}$.

That is, $$ B = \pm R\sin\frac{\ell_{1}}{2R},\qquad C = \pm R\sin\frac{\ell_{2}}{2R}, \tag{2} $$ and consequently $$ \frac{B}{\sqrt{R^{2} - B^{2}}} = \pm\tan\frac{\ell_{1}}{2R},\qquad \frac{C}{\sqrt{R^{2} - C^{2}}} = \pm\tan\frac{\ell_{1}}{2R}. \tag{3} $$

Substituting (3) into (1), $$ \cos\theta = \frac{B}{\sqrt{R^{2} - B^{2}}}\, \frac{C}{\sqrt{R^{2} - C^{2}}} = \pm \tan\frac{\ell_{1}}{2R} \tan\frac{\ell_{2}}{2R}. $$ To get the large angle, either take the arccos of the negative value, or $\pi$ minus the arccos of the positive value. (The angle/area formulas above do the latter.)

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    $\begingroup$ @HarishChandraRajpoot: You will need to know at least two of the angles of your quadrilateral to find the area, I think. $\endgroup$ Mar 25, 2015 at 13:31
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    $\begingroup$ I agree with @Kundor, I think. Consider analogous shapes in the plane. If all you know about the shapes in the plane is that two opposite sides have length $l$ and the other two have length $b,$ but not the angles, you cannot calculate the area. I think a more reasonable definition of spherical rectangle would, as user86418 says, require all four angles equal. $\endgroup$
    – Randy E
    Mar 25, 2015 at 13:39
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    $\begingroup$ Yes, all four interior angles are equal in magnitude (but each is greater than 90 degree) $\endgroup$ Mar 25, 2015 at 13:41
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    $\begingroup$ So then the area is $r^2(4\theta-2\pi)$ if you know $\theta$. I feel like it should be possible to calculate $\theta$ based on $b$, $l$, and $r.$ $\endgroup$
    – Randy E
    Mar 25, 2015 at 13:48
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    $\begingroup$ (+1) I am impressed that you arrived at the correct answer doing this with Cartesian coordinates, though the answer can be simplified using the identity $\frac\pi2-\cos^{-1}(\theta)=\sin^{-1}(\theta)$. However, using spherical trig makes things a lot simpler. $\endgroup$
    – robjohn
    Apr 5, 2015 at 9:59
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Let the north-pole $N=(0,0,1)$ be the center of the rectangle, and the planes $x_1=0$, $\>x_2=0$ be its symmetry planes. Let the point $$P=(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)$$ with $0<\phi<{\pi\over2}$, $\>0<\theta<\pi$ ($\theta=0$ at the north pole!) be one of its vertices. Then we see two rightangled spherical triangles, both with hypotenuse $NP$ of spherical length $\theta$, one of them with angle $\phi$ at $N$ and the other with angle ${\pi\over2}-\phi$ at $N$. The legs opposite these angles have spherical lengths ${a\over2}$ and ${b\over2}$ respectively, where $a$ and $b$ were given in advance.

Napier's rule for rightangled spherical triangles then give $$\sin{a\over2}=\sin\phi\sin\theta,\quad \sin{b\over2}=\cos\phi\sin\theta\ .$$ From these two equations one can compute $\theta$ and $\phi$, and other Napier's formulas then give the missing angles at $P$. Knowing all angles one obtains the area at once; see user86418's answer.

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I believe it follows from this formula for spherical area of quadrangles on Wikipedia that the area should be $$ 4 \arctan\left(\sin\left(\frac b 2\right) \tan\left(\frac \lambda 2\right)\right), $$ assuming that the radius of the sphere is 1, where $$ \lambda = \arccos\left(\frac{\cos(a) - \sin\left(\frac b 2\right)^2}{\cos\left(\frac b 2\right)^2}\right). $$ For this, we take an "equator" (a great circle) bisecting the quadrilateral through the sides of length $b$. We now have two congruent quadrilaterals; each has two right angles at the equator, and the other two angles are some $\gamma > \frac {\pi}{2}$. The Wikipedia formula for the area $E_4$ of these quadrilaterals should apply, with $\phi_1 = \phi_2 = \frac b 2$, $\lambda_1 = 0$, and $\lambda_2$ is the difference in longitude of an arc of length $a$ beginning and ending at latitude $\frac{b}{2}$.

To determine this, we use the formula for great-circle distance between points, and solve for the longitude difference $\Delta \lambda$.

For a sphere of radius $R$, divide every appearance of $a$ and $b$ by $R$, then multiply the result by $R^2$.

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For a quick approximation, I recommend this simple approach (modified to remove degree conversion): link.

$$A=R^2\left(\sin(\text{polar angle 1})-\sin(\text{polar angle1})\right)\left(\text{azimuthal angle 1}-\text{azimuthal angle 2}\right)$$

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    $\begingroup$ Effectively, you're just integrating cos(latitude) for the latitudes in question and then taking the appropriate portion of the result based on bounding longitudes. $\endgroup$
    – user2469
    Dec 19, 2016 at 17:24
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    $\begingroup$ Correct, it's a quick approximation and should produce different results between the equator and the poles, as it does not account for great circle arcs. $\endgroup$ Dec 24, 2016 at 23:38
  • $\begingroup$ Oh, actually I was saying that it WAS completely accurate, just a much simpler way to get the same answer. $\endgroup$
    – user2469
    Dec 25, 2016 at 19:54
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The solid angle subtended by any spherical rectangle with length $l$ & width $b$ each as a great circle arc on a spherical surface of radius $R$ is given by following generalized formula as follows $$\omega=4\sin^{-1}\left(\tan \left(\frac{l}{2R}\right)\tan \left(\frac{b}{2R}\right)\right)$$

hence the area ($A$) of spherical rectangle will be given as follows

$$A=\omega R^2$$ $$\boxed{\color{blue}{A=4R^2\sin^{-1}\left(\tan \left(\frac{l}{2R}\right)\tan \left(\frac{b}{2R}\right)\right)}}$$

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    $\begingroup$ This is the same as my answer, scaling for radius $R$ instead of $1$. $\endgroup$
    – robjohn
    May 2, 2020 at 6:48
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    $\begingroup$ OK, but in your results, radius $R$ is missing from internal terms like $\tan(l/2)$ should be written as $\tan(l/2R)$, I mean how to scale internal terms? $\endgroup$ May 2, 2020 at 6:50
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    $\begingroup$ As I said, my sphere has radius $1$, so there would be no $R$. $\endgroup$
    – robjohn
    May 2, 2020 at 6:51
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    $\begingroup$ I think it should be generalized for any radius $R$ as radius 1 is a specific case $\endgroup$ May 2, 2020 at 6:52
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    $\begingroup$ It is trivial to scale for radius. I have now done so in my answer. $\endgroup$
    – robjohn
    May 2, 2020 at 6:56
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area of any n-polygon lying on the sphere which has radius r will be equal to

((Sum of interior angles)-$(n-2)\pi$)$(r)^2$

otherwise,

using Solid Angle concept

=(solid angle subtended by rectangle at the center of sphere) x(radius)$^2$

Hope this helps

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    $\begingroup$ how do you find solid angle? can you elaborate your answer? $\endgroup$ Apr 21, 2021 at 7:16

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