What’s Your Angle

Now that we have our measurements it’s time to do some math. We need only one formula:


The law of sines states that for any planar triangle the sin of an angle divided by its opposite side is always equal to the other two ratios. This comes in handy when you know any two angles and the length of a side (or the length of two sides and one angle).

Referring back to our original diagram and our measurements we know:

  • R = 1.00 cm
  • r = 0.660 cm
  • R-r = 0.340 cm
  • S = 16.8 cm (by assumption)

We have two sides of the triangle bounded by L’, R-r, and S and know that the angle opposite S is 90 degrees. Since we know the length of the side opposite angle e we can use the law of sines to calculate it.


Now we can start talking about the cone that makes up the mouthpiece! Referring back to the original diagram we can note the following from basic geometry:

  1. L’ and line segment Ss (the side of the cone) cross making e and e’ vertical angles. Vertical angels are always equal.
  2. L’ is a construct which is parallel to L which means it is perpendicular to r. Therefore angle b is 90-e’ or 90-e (see above).
  3. The sum of the interior angles of a planar triangle always equal 180 degrees. Angle c is 180-a-(b-e’) which is 180-90-(90-e’) or e’.
  4. Another way to determine angle c is is to note that if you extend the line segment Ll (running through the center of the cone) and L’, these two parallel lines are crossed by the line segment Ss (the side of the cone). In this case e’ and c are alternate interior angles which are always equal.

Now we have enough information to calculate the volume of our theoretical mouthpiece.

Next: How Big?

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