Formula for the Arc Length of a Helix
A helix, also known as a cylindrical coil or solenoid coil, is a three-dimensional spiral curve generated by tracing a circle of constant radius at a constant rate while simultaneously increasing the height at a constant rate. Put another way, a helix wraps around an imaginary cylinder in such a way that the space between consecutive loops remains constant. Examples of helices include
- threads of a screw
- spines of DNA molecule (double helix)
- simple coil springs in clickable pens
- string wrapped around a tube
- ringlets of hair
One can easily model a helix with a cylindrical can and a rectangular piece of paper that exactly fits all the way around the can. Simply draw a diagonal line across the rectangle and wrap it around the can -- voilà, you have just created one full turn or loop of a helical spiral.
You can calculate the arc length of a helical spiral if you know its diameter, number of full turns (loops), and the total height of the helix. You can also calculate the arc length of a helix from its parametric equation if you know it. Both methods are described below.
Basic Arc Length Formula for a Helix
The arc length of a helix is the length of one full turn times the number of turns (which may be fractional). If the diameter of the coil is D, the total height H, and the number of turns T, then the arc length of the helix (aka cylindrical spiral) is given by the expression
= T*sqrt[ (πD)^2 + (H/T)^2 ]
= sqrt[ (πDT)^2 + H^2 ]
The part of the formula "sqrt[ (πD)^2 + (H/T)^2 ]" is the length of the diagonal of a rectangle whose height is H/T and whose width is πD. This is the size of the rectangle that wraps around one full turn of the helix whose diameter is D. This distance is multiplied by "T" in the equation because the length is counted once for each full turn of the spiral. This is shown in the image below.
Arc Length in Parametric Coordinates
Helices can be parameterized by a set of equations such as
x(t) = R*cos(t)
y(t) = R*sin(t)
z(t) = C*t
where R is the radius of the helix (D/2) and C is a constant such that 2πC equals the height of one full turn. If the parameter t ranges from 0 to K, then the number of turns is K/(2π). The equations for x(t), y(t), and z(t) can be switch to yield a helix with a different orientation. The arc length of a helix described in parametric coordinate notation is given by the integral
∫ sqrt[ x'(t)^2 + y'(t)^2+ z'(t)^2 ] dt
from t = 0 to K. Since x'(t) = -R*sin(t), y'(t) = R*cos(t), and z'(t) = C, the integral simplifies to
∫ sqrt( R^2 + C^2) dt, 0≤t≤K
= K*sqrt(R^2 + C^2)
We can show that this formula is consistent with the first formula for the arc length of a cylindrical spiral using the following equivalences:
R = D/2
K = 2πT
C = H/(2πT)
Then the integral formula becomes
K*sqrt( R^2 + C^2)
= 2πT*sqrt[ (D/2)^2 + H^2/(2πT)^2 ]
= T*sqrt[ (2πD)^2 + (H/T)^2 ]
Radius of Curvature and Torsion in a Helix
For a helix defined by the parametric equations x(t) = R*cos(t), y(t) = R*sin(t), and z(t) = C*t, the curvature and torsion of the helix are given by the equations
curvature = R/(R^2 + C^2)
torsion = C/(R^2 + C^2)
A metal spring has a diameter of 23mm. The total height of the spring is 134mm and the number of turns the spiral makes is 12.5. Using the values D = 23, H = 134, and T = 12.5, we calculate the arc length as
12.5*sqrt[ (23π)^2 + (134/12.5)^2 ]
= 12.5*sqrt[ 529π^2 + 10.72^2 ]
= 12.5 * 73.0475
The total length of the coil is therefore 913.09 mm, or 91.309 cm, or 0.91309 m.
A helix is parameterized by the vector equations
x(t) = (6/π)*sin(t)
y(t) = (6/π)*cos(t)
z(t) = (2.5/π)*t
What is the arc length from t = 0 to t = 6π? In this problem we have R = 6/π, C = 2.5/π, and K = 6π. The number of full loops of the spiral is 3. Plugging the values of R, C, and K into the formula K*sqrt(R^2 + C^2) gives us
arc length = 6π*sqrt( 6.25/π^2 + 36/π^2 )
= 6π*sqrt( 42.25/π^2 )
= 6π * 6.5/π
Therefore the arc length is exactly 39. Its curvature and torsion are
curvature = (2.5/π)/(42.25/π^2) = 10π/169
torsion = (6/π)/(42.25/π^2) = 24π/169
Road Bait and Akorn are two squirrel friends who like to chase each other up and around a wooden utility pole. The pole is 15 inches in diameter and 31 feet tall. They start at the base and run in a helical path making 9.75 rotations. When they reach the top, how far have they run?
In this problem, D = 15 inches = 1.25 feet, H = 31 feet, and T = 9.75. Plugging these values into the formula gives us
arc length = sqrt[ (πDT)^2 + H^2 ]
= sqrt[ (12.1875π)^2 + 31^2 ]
= sqrt[ 2426.98323184 ]
Therefore, the squirrels run about 49.2 feet. If they start over, how many rotations must they make around the pole so that the total distance they run is 100 feet? To solve this problem, we plug in the known values and solve the following equation for T:
100 = sqrt[ (1.25πT)^2 + 31^2 ]
This gives us
10000 = (1.25πT)^2 + 961
9039 = (1.25πT)^2
95.0736556571 = 1.25πT
95.0736556571/(1.25π) = T
T = 24.2103
So the squirrels must make about 24.2 loops around the pole to achieve a distance of 100 feet.
What length of wire do you need to make a helix that has a diameter of 3 inches, 4 full loops, and a vertical gap of 1 inch between the coils?
To solve this problem, we calculate the length of wire needed to make one loop of the helical spiral with a height of 1 inch and then multiply it by 4. The length wire needed for one loop is
sqrt[ (3π)^2 + (1)^2 ]
= sqrt[ 9π^2 + 1 ]
= 9.4777 inches.
Therefore, the total amount of wire needed is 4*9.4777 = 37.91 inches, or about 3 feet and 1 29/32 inches.
String is wrapped evenly around a tube 5 times. If the length of the string is 65.7 cm and the length of the tube is 29.3 cm, what is the diameter of the tube?
For this problem we have T = 5, L = 65.7, and H = 29.3. Using the helix length formula we have
65.7 = 5*sqrt[ (Dπ)^2 + (29.3/5)^2 ]
Solving this for D gives us
13.14 = sqrt[ (Dπ)^2 + (5.86)^2 ]
172.6596 = (Dπ)^2 + 34.3396
138.32 = (Dπ)^2
sqrt[ 138.32 / π^2 ] = D
D ≈ 3.74 cm