1
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ You have an error due to some confusion: $b+2\pi R/X$ (in radians). $\endgroup$
    – Bernard
    Oct 2 '19 at 0:14
2
$\begingroup$

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 $2\pi$:

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

$\endgroup$
3
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.