# Angle relationship between cone and its surface

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.

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?

• You have an error due to some confusion: $b+2\pi R/X$ (in radians). 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 $$2\pi$$:

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

• 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. Oct 2 '19 at 1:11
• 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. Oct 2 '19 at 1:17
• You're right, the last line of your answer is the "simple" relationship between the angles. That does answer the question. Oct 2 '19 at 2:25