A cylindrical spiral is more commonly called a helix. A Pythagorean relationship of certain segments of the cylinder (real or imagined) about which the helix spirals may be used to calculate the length of the helix.
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Orient The Helix
The primary component of the helix's coordinate system is the cylinder about which the helix spirals. Draw this object. The perimeter of the circular plane will be used as a proportional. Since this length depends only on the length of the radius (P = 2pi(Radius)) of the circular plane, draw the radius and label it as "R." The other proportional that is needed is the length along the major axis of the cylinder that measures one full turn of the helix. Identify this and label it as "H."
Draw the Proportional Triangle
The length L of one full turn of the helix shall be the hypotenuse of a right triangle where the shorter dimensions shall be given by H and the perimeter of the circular plane of the cylinder (2piR). To visualise the proportion, imagine that the triangle is wrapped about the surface of the cylinder fully attached along the perimeter. Draw a triangle and label the hypotenuse of the triangle as L. The shortest side of the triangle shall be H and the remaining side represents the perimeter, 2piR.
Determine the Proportion
The right triangle from Step 2 allows for the use of the Pythagorean theorem. So, write the relationship L = sqrt(H^2 + (2piR)^2) where "sqrt" means "take the square root." This shall give the length of just one full turn on the helix. The full length of the helix can be determined by scaling the full length of the cylinder's major axis by the proportion L/H = sqrt(1 + 4pi^2(R/H)^2). So, if the cylinder whose major axis spans 100 inches with a radius of 1 inch and H = 5 inches, then L/H = sqrt(1+4pi^2(1/5)^2) = 1.61, and the total length is 1.61(100 inches) = 161 inches.
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