The question is as follows:

Both the slant height and the base diameter of a cone are 12 inches. What is distace between two opposite points on base circle of cone, if it is required that the path must lie on the lateral surface of the cone?

I am not sure of where to start with this problem, therefore, any help with how to start this will be very much appreciated. Thank you in advance.

  • $\begingroup$ Are you looking for the shortest path across the curved surface? $\endgroup$ – David Quinn Feb 11 '18 at 17:39
  • $\begingroup$ "How far is it .. " , it = what ? $\endgroup$ – G Cab Feb 11 '18 at 18:37
  • $\begingroup$ @geo_freak Hope edit is OK. Else please restore the same. $\endgroup$ – Narasimham Feb 11 '18 at 18:48

The answer is best seen on a development.

Semi-vertcal angle is $\sin^{-1}\frac12 = 30^{\circ}$

On development angle subtended at cone apex is

$$ \frac12* 360^{\circ}=180^{\circ}$$

If $l=2r = 12 inches, \, $say, cone develpment is a semi-circle of r= 6 units radius.

Minimum (geodesic) distance is shown by red line $$ = r \sqrt 2 \, or \, 6\sqrt 2 $$

enter image description here

  • $\begingroup$ How have you found out the semi-vertical angle is $\sin^{-1}\frac12 = 30^{\circ}$? What does the semi-vertical angle mean in this context? $\endgroup$ – geo_freak Feb 11 '18 at 19:01
  • $\begingroup$ $ \sin \alpha = \dfrac{r}{l} (given) = \dfrac12$ This helps to draw the development. $\endgroup$ – Narasimham Feb 11 '18 at 19:12

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