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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.

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  • $\begingroup$ Are you looking for the shortest path across the curved surface? $\endgroup$ Feb 11, 2018 at 17:39
  • $\begingroup$ "How far is it .. " , it = what ? $\endgroup$
    – G Cab
    Feb 11, 2018 at 18:37
  • $\begingroup$ @geo_freak Hope edit is OK. Else please restore the same. $\endgroup$
    – Narasimham
    Feb 11, 2018 at 18:48

2 Answers 2

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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

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  • $\begingroup$ 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, 2018 at 19:01
  • $\begingroup$ $ \sin \alpha = \dfrac{r}{l} (given) = \dfrac12$ This helps to draw the development. $\endgroup$
    – Narasimham
    Feb 11, 2018 at 19:12
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I came upon this math question while looking for something else. The question is a complicated way (perhaps meant to test one’s understanding of the terms) of addressing the very fundamental geometry of the Pythagorean theory: A squared plus B squared equals C squared-the sides A and B being the horizontal and vertical sides of a right triangle, and side C being the diagonal of the triangle. Never mind the cone, what we’re looking at is a simple right triangle. And since the question tells us the height and width are both 12”-actually we don’t even need to know the height in this case- the answer is double the base width, or 24”. It’s a very simple question.

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  • $\begingroup$ And don’t let the ‘slant height’ confuse you. The question only addresses the height of the slant, not the running distance of the diagonal, which happens to be 16.970 “. Study the Pythagorean theory, and it will all become very obvious. $\endgroup$
    – user885474
    Feb 9, 2021 at 22:58
  • $\begingroup$ If the question is asking us to travel along the circular base of the cone to the opposite side, then we already know the radius is 12”. So the diameter is 24”. Using the pi r squared theory, multiply 3.14 (pi) times 24” which equals 75.36 (our circumference). Half that circumference will take us to the opposite side and travel 37.68”. $\endgroup$
    – user885474
    Feb 9, 2021 at 23:42

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