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.
Length of straight line between A and C = radius of the
circle * 1.556985
[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 "