For the past day, I have been trying to calculate the arc length of a sphere from a pole to a longitudinal cross-section with a specified circumference.

I have diagrams and equations belo w. However, I am getting answers that seem incorrect. Are equations correct? Any improvements to it?

enter image description here

  • $\begingroup$ Why the down vote? I have shown my work and how I got to the equation $\endgroup$ – Christopher Rucinski May 14 '15 at 21:33
  • $\begingroup$ See math notation guide. $\endgroup$ – user147263 May 15 '15 at 0:47

Computing the distance $d$ using $\theta$ in degrees is...ill-advised. :) Radians are defined so that $d = r_{0}\theta$; mathematics seldom gets simpler than that. Further, if you're using a calculator to test values, there's a non-negligible chance the outputs are coming to you in radians.

That aside, you're essentially there. If you measure $\theta$ in radians, then \begin{gather*} C = 2\pi r_{i},\quad\text{or}\quad r_{i} = C/(2\pi); \tag{1} \\ \sin\theta = r_{i}/r_{0},\quad\text{or}\quad \theta = \arcsin(r_{i}/r_{0}); \tag{2} \\ d = r_{0}\theta = r_{0} \arcsin\bigl[C/(2\pi r_{0})\bigr]. \tag{3} \end{gather*} For example, if $r_{0} = 1$, the equator has $C = 2\pi$, and the formula gives $d = \pi/2$ as expected. If $C = \pi$, then $d = \pi/6$ or $5\pi/6$, again as expected.

| cite | improve this answer | |
  • $\begingroup$ if $r_0 = 1km$, then $d =( \pi / 2 )km$ ? $\endgroup$ – Christopher Rucinski May 14 '15 at 22:54
  • $\begingroup$ Indeed, the calculator was set to radians when I was using degrees $\endgroup$ – Christopher Rucinski May 14 '15 at 23:08
  • $\begingroup$ If $r_{0} = 1$ km, you still need a circumference to find $d$. :) (In case it matters, trigonometric formulas in calculus hold only when angles are measured in radians; nature prefers radians as the unit of angle.) $\endgroup$ – Andrew D. Hwang May 14 '15 at 23:29
  • $\begingroup$ Yes, I got it to work with degress, but I will convert it to use radians instead. Also, my km question was in reference to your example ..... if $r_0=1$km, the equator has $C=2π$km, and the formula gives $d=π/2$km as expected. I just wanted to make sure i didn't have to do anything else with my answer - I don't have to I think $\endgroup$ – Christopher Rucinski May 14 '15 at 23:34
  • $\begingroup$ Ah...yes, $\pi/2$ km. :) $\endgroup$ – Andrew D. Hwang May 14 '15 at 23:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.