# 33 Degree Dual Cone Resonator

(Windows has a calculator capable of sin and tan functions)
(sin = sine   tan = tangent on the calculator pad)
(These are Trigonometry Functions) ### Diameter = Height * .592426989924160

(1/2 Diameter)/Height  = Tangent (16.5 deg)
Diameter =  2 * Tan (16.5) * Height
Diameter = Height *  0.59242698992416048848841940930336

To use a calculator instead of typing all those numbers, type in the following string
16.5  tan   *  2  * [Type in the height]   =
You now have the diameter to 32 places or more accuracy
Hit   M+  key to store this in the memory for later retrieval

### Side Length = Cone Base Diameter * 1.7604682610

1/2 Diameter / Side Length = Sine (16.5 Deg)
Side Length =  (Diameter / 2)* Sine (16.5)
Side Length = Diameter * 1.7604682610414406827257050702004

Calculator entry string
16.5  sin   *  MR [key to recall the diameter]  / 2   =
You now have the side length to a very accurate length

### Side Length = Height * 1.04294891274580 A cone can be constructed by cutting a wedge out of a circular piece of material.
The Radius of the circle will become the side length of the rolled up cone.
Points A and C must be plotted on the circle, to know where the overlap position is located, shown on the right now touching.

It can also be helpful to know the straight line distance between points A and C for plotting them as a triangle on the circle of the material.
This is more accurate then using a protractor to measure the 102.245 degree angle, however both should intersect the same point on the curve.

### [For those who want to confirm the math and geometry the following proof is offered] The Blue wedge will be cut out of the circle and this piece rolled into a cone of 33 degree angle.
The left over white piece can be used to cut a second cone, and then a third if desired.
This will depend on overlaps if added along the sides.

Polygon Formula for Segment size on a circle

Fractal Segment Length = Diameter (sin (1/2 the angle))
Length AB and Length AC can be found using this formula if the diameter of the circle and the angles are known.

Length AB = Diameter (sin 16.5 deg)
Length AC = Diameter (sin (1/2 the angle in deg))

The first function to notice is the line AB is the diameter of  the 33 deg cones base, and this is a circle, that will have circumference of the outer curve along ABC when the material is rolled up.
The distance of that circumference of the cones base is distance (AB * Pi) and is equal to the perimeter of the circle between points A and C.

Perimeter AC =  Pi * Length AB

Now from the above polygon formula substitute Length AB as Diameter (sin 16.5 deg)

Perimeter AC = Pi * Diameter * (sin 16.5 Deg)

Now notice that [Pi * Diameter] is the circumference of the circle at 360 degrees.

Perimeter AC = Circumference *  (sin 16.5 deg)

And that there is another important ratio to observe
Perimeter AC / Circumference = Angle AC / 360 Deg
This transforms
Angle AC = Perimeter AC * 360/ Circumference

Now substitute from above [Perimeter AC = Circumference *  (sin 16.5 deg)]

Angle AC = Circumference * (sin 16.5 deg) * 360 / Circumference

Circumference cancels to 1

Angle AC = 360 * (sin 16.5 deg)

### Angle AC =  102.24552409341214227998028865171

Length AC = Diameter (sin(1/2 102.24..... deg))
Length AC = 2 * Radius * (sin 51.122762046706071139990144325853 deg)

### Length AC = Radius * 1.5569851193202311401894419374101

[Long Proof, Geometric Formula, Trigonometry, 33 degree cone]

### Worksheet

The Following work sheet was provided for quick calculations once you have decided the height of your cone.
Bashar indirectly suggested not smaller then 61.6 cm, or 24.3 "

Mark the units you use for reference [Inches or Centimeters or Millimeters]

Inch ____ cm ____ mm ____

Start by choosing the height of the cone you want to create.

Cone Height ________________________

In a 33 degree cone, there is a constant ratio between the height of the
cone and the diameter of the base circle. That ratio is .59242698

Diameter = Height * .59242698

Cone Base Diameter _____________________

Next find the side length of the cone SL.
SL = diameter of the cone / 2*(sine 16.5 deg)

Side Length = Cone Base Diameter * 1.7604682610

Cone Side Length ___________________ (Radius of the Circular sheet)

Now refer to the Graphic and this length becomes the radius of the large
circle you will be constructing to become the surface of the cone.

Draw a circle with radius = Cone Side Length from above calc, on your flat material.

Draw the radius in, to intersect the circle at point a.
Now either use a protractor to draw the second line at 102.245 degrees
to point c, or calculate the line a - c length and measure it along the
curve of the circle to mark the point.

Length of straight line a - c = radius of the circle * 1.556985

a to c straight line distance ____________________

If you want an overlap flap, mark that out also before cutting the pattern.
Be sure the lines are clearly visible for folding the cone so the points a
and c will touch. The cone should then be a perfect 33 degree angle at the
tip to the bottom.

A disc can be cut for rolling the bottom of the cone into its final circle.
Its diameter is already calculated above. The first calculation.

This work sheet can be used to quickly create cones of any height desired.

If you find a mistake in the math, please send feedback for correction.

Thanks,
Dave L