Say I start with the cone shown in the left in the diagram below. I can find the angle a formed by the wall of the cone as 2*arcsin(R/S).

If I cut the cone open and flatten the wall into a 2D surface, it forms a segment of a circle having radius S and an arc length equal to the circumference of the cone, 2πR. The segment angle b can be found as the fraction of the circle comparing the segment arc to the circle circumference, which reduces to 360*R/S.

enter image description here

If I want to calculate the actual angles, I need a trig table for the cone. If I'm only interested in the relationship between a and b, I could calculate the angles and compare them. However, it intuitively seems like there might be some simple ratio or relationship between the two angles.

That's my question. Is there a simple ratio or relationship between the cone angle and the angle of its flattened surface?

  • $\begingroup$ You have an error due to some confusion: $b+2\pi R/X$ (in radians). $\endgroup$
    – Bernard
    Oct 2 '19 at 0:14

I'll do this first with all angles in radians, since the relationships are simpler that way. Therefore from your formula for $b,$ but using $2\pi$ radians rather than $360$ degrees, we have $$ b = \frac{2\pi R}{S}. \tag1$$

Therefore, after dividing both sides of Equation $(1)$ by $2\pi$, $$ \frac RS = \frac{b}{2\pi}. \tag2$$

You also found that $$ a = 2 \arcsin\left(\frac RS \right). \tag3$$

Now use Equation $(2)$ to substitute for $\frac RS$ in Equation $(3)$: $$ a = 2 \arcsin\left(\frac{b}{2\pi} \right) = 2 \arcsin\left(\frac{b}{360^\circ} \right). \tag4$$

The version with division by $360^\circ$ is in case you insist on measuring $b$ in degrees and don't want to convert it to radians.

To get $b$ in terms of $a$ we just undo all the things we had to do to $b$ in order to get $a$, starting with the multiplication by $2.$ That is, divide by $2,$ take the sine, and multiply by $2\pi$:

$$ b = 2\pi \sin\left(\frac a2 \right) = 360^\circ \times \sin\left(\frac a2 \right). $$

  • $\begingroup$ Thanks for your response. I had already played with these kinds of equivalents. My question was really if a/b is a constant or some simple relationship that doesn't require trig tables. I realized this isn't the case. I'll post an answer to close the loop. $\endgroup$
    – fixer1234
    Oct 2 '19 at 1:11
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    $\begingroup$ The gist of the answer is that this is as simple as it gets. If it could be any simpler, you could simplify all trig calculations in a similar way, because you would have discovered a "simpler" formula for the sine function. $\endgroup$
    – David K
    Oct 2 '19 at 1:17
  • $\begingroup$ You're right, the last line of your answer is the "simple" relationship between the angles. That does answer the question. $\endgroup$
    – fixer1234
    Oct 2 '19 at 2:25

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