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There are two conical sheets inside a sphere of radius r. The cones are hollow and exactly same. The spherical surface is cut at two places by the circular portion of the cones. How to calculate area of a part of the spherical surface excluding the parts cut by the cones as shown in the image? The cones subtend an angle of $21^°$ with their axis as shown.Height of each cone is equal to radius of the spherical surface.

enter image description here

$$Fig.1 - Two\ dimensional\ view\ of\ the\ spherical\ surface.$$

Is there any measure like a solid angle (in 3D) which can be estimated for the part of the spherical surface compared to a full cylindrical surface with area $4πr^2$ and thus the area of part of the spherical surface excluding the cones can be calculated ? Please suggest any techniques.

Reference: Engineering electromagnetics ,Hayt and buck, 8e, Mcgrawhill india, 2014, Fig.9-18,page 268

Amit M
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    Why not calculate the surface area of the two caps, say using spherical coordinates, and subtract that from the surface area of the sphere? – Michael Burr Jan 09 '25 at 10:58
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    This gives a clue. Think of excluding the polar regions. – Oscar Lanzi Jan 09 '25 at 11:40
  • Michael Burr, That could work. I will try to calculate the area of the two curved parts of spherical surface on top of the cones using spherical coordinates. – Amit M Jan 09 '25 at 11:45
  • If the height of each cone is equal to the sphere's radius then the circular base is tangent to the sphere and outside the sphere except at the point of tangency. I suppose it doesn't really matter how large the cones are as long as they are large enough to intersect the sphere and cut out the two spherical caps. – David K Jan 09 '25 at 14:54
  • I'd like to say this is a duplicate question, but there are so many ways to do the problem that I don't know which one to pick! See these questions: https://math.stackexchange.com/search?q=spherical+cap+area – David K Jan 09 '25 at 15:04
  • Why not using steradian definition and formula. see https://de.wikipedia.org/wiki/Raumwinkel – sirous Jan 09 '25 at 15:12
  • sirous, I got the same result for the area of curved top surface of the cone as this: Raumwinkel eines Kegels – Amit M Jan 09 '25 at 16:31

2 Answers2

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The projection from the sphere to the cylinder is area-preserving (Showing the projection from the sphere to the cylinder is area preserving). Thus, to calculate the surface area of the sphere without the polar regions cut out by the cones, comes down to calculating the surface area of a cylinder with radius $r$ and height $2h$ where $h$ is the height of one of the cones. In the special case where $h = r$ we find back the full surface area of the sphere $2 \pi r \cdot 2r = 4\pi r^2$.

Wannes
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You can calculate the area using spherical coordinates. By definition $\phi$ is the angle with the z-axis. You get the following integral \begin{gather*} \int_0^{2\pi} d\theta \int_{\frac{21}{360}2\pi}^{\pi - \frac{21}{360}2\pi} r^2 sin\phi \; d\phi \approx 11.7317 r^2 \end{gather*}

Sam
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  • It may depend on how you view the axes. θ is the angle with z-axis. ϕ is the angle along y-axis .This is if you see the origin straight before you and z axis vertically upward, x-axis towadrs you and y-axis towards your right. The cone extends for ϕ = 0 to ϕ = 2π degrees/radians. Its just that its θ is limited. So the definite integral should have limits from 0 to 2π for the integrand dϕ ? – Amit M Jan 09 '25 at 16:20
  • You seem to have calculated the lateral area of the spherical surface. – Amit M Jan 09 '25 at 16:36