Calculate a rotary cone opening angle

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All cones of rotation have an opening angle at their upper end, which can be regarded as a measure of how acute or wide the cone of rotation is. This angle can easily be calculated from the given sizes of the cone.

Opening angle of the rotary cone - you should know that

  • First of all, get an idea of ​​how such a rotating cone could have come about: Take your set square (or alternatively any one other triangle) and place it upright on the table so that one side of the triangle lies flat on the table top (not the Top!).
  • Now turn this triangle with the side lying on it once around the upright side of the triangle - the result is a cone of rotation as an imaginary figure.
  • This cone of rotation is more acute, the smaller the angle is at its tip and the wider, the larger this angle is. Ultimately, the upper angle of the rotated triangle determines the opening of the cone. This angle is therefore called the opening angle.

How to calculate the opening angle

After you have made it clear to yourself that there is actually a triangle in every rotary cone, the important sizes of every cone result from this triangle, namely the surface length s (which is on the outside), the radius r of the cone (which corresponds to the rotating side of the triangle and thus the radius of the circle) and the height h of the cone (i.e. the distance between the circle and Top).

  • The opening angle (it is called alpha) is now in the upper tip of the cone triangle, namely between the surface length s and height h. Alpha is opposite the radius r.
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  • Depending on which of the other sizes s, h or r you have given, you can use the trigonometric Functions calculate the opening angle alpha. Two of these three sizes are always required.
  • Given h and r, calculate tan (alpha) = r / h
  • Given h and s, calculate cos (alpha) = h / s
  • Given r and s, calculate are (alpha) = r / s.
  • Alpha results with the calculatorby inverting the trigonometric functions (SIN-1 resp. ARCSIN, depending on the model.

Opening angle - a calculated example

You should calculate the opening angle for a cone with r = 3 cm and h = 5 cm.

  1. In this case you need the tangent function (see above).
  2. The following applies: tan (alpha) = 3/5 = 0.6
  3. With the inverse tangent function you get: alpha = 30.96 °.

Note: The opening angle is often not angle alpha, but its double, in this case around 62 °.

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