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The question was: Sketch the intersection of a sphere and a plane that does not pass through the center of the sphere. If you know the circumference of the circle formed by the intersection, can you find the surface area of the sphere?

Of course, this question is searching for a theoretical method and solution.

My first thought was to set corresponding variables and allow the cross-section that contains the circle to be parallel to the plane that intersects the diameter of the sphere - or the great circle. Of course, this creates two circles that lie on parallel planes, and therefore we can construct several right triangles, and resort to Trigonometry to solve this problem.

I set an angle θ that is a central angle, and increases as the distance between the origin of the great circle and the cross-sectional circle decreases, and cannot fully reach a half rotation of 180 degrees. I then connected the collinear points that form the radius of the sphere, which consist of the center of the sphere, the center of the cross-sectional circle, and the endpoint of the radius. I then formed a kite by connecting the endpoint of the radius with the endpoints of the diameter of the cross-sectional circle.

Hopefully the above describes the graphics I created.

How do I do this? How do I prove or disprove that the surface area is determinable given the circumference of a cross-sectional circle that lies on a different plane?

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2 Answers 2

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The sphere can be represented as the points $(x,y,z) \in \mathbb{R}^3$ such that $x^2 + y^2 + z^2 = R^2$ . Then, assuming the plane is horizontal(it can be rotated by a change in coordinates because the sphere is symetrical in all dimensions) we can express the plane as ${z = k}$ for some $k \in \mathbb{R}$, which would be the distance of the plane to the origin. Then the intersection of the sphere with the plane is ${x^2 + y^2 = R^2 - k^2}$ For it to be a circle, $-R < k < R$. Because we are given the circumference of this circle(lets call it $c$) we have that $c = 2\pi\sqrt{R^2 - k^2}$. Then it is easy to get the value of $R$ with respect to $c$ and because the sphere is fully determined by its radius we can also get its surface area from $c$.

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  • $\begingroup$ This makes complete sense, but how do we determine the value of k if R is unknown? $\endgroup$
    – Tricryo
    Dec 7, 2020 at 20:07
  • $\begingroup$ @Tricryo You need the value of $k$ to get $R$, otherwise for every $R$ you can pick some $k\in\mathbb{R} : -R < k < R$ so that the intersection has the circumference that you want, so the problem wouldn't be determined. The other answer explains this, in mine I assumed the plane equation was given. $\endgroup$
    – Bolito2
    Dec 8, 2020 at 23:52
  • $\begingroup$ So you're saying that in order to solve this problem, an auxiliary value k is required. Without k, the surface area is unsolvable? I think that's the case $\endgroup$
    – Tricryo
    Dec 10, 2020 at 7:48
  • $\begingroup$ Yes, if the plane equation isn't given(it actually suffices with its distance to the origin, what I called $k$) there is no solution. $\endgroup$
    – Bolito2
    Dec 10, 2020 at 22:08
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Imagine a big sphere and several planes going through it at various distances from the sphere center. Some intersections will have smaller circumference than others (start with a plane nearly tangent to sphere). So not determined.

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