The surface area of a coil is calculated in a similar manner to the surface area of a cylinder. Mathematically speaking, cylinders are stacks of circles and so their surface area can be derived from the sum of the circumferences of these circles. As the circumference of a circle is 2πr, the cylinder surface area must be 2πrl, where “r” is the radius of the circle and “l” is the length of the cylinder.
How to Calculate the Circumference of a Circle
Begin by measuring the diameter of the circle at one end of the coil. This is done by measuring the distance across the end of the coil. Because the diameter is twice the length of the radius, the equation becomes circumference = π times diameter; meaning it is not necessary to then calculate the radius in order to measure the circumference (Furlonge, Mathematics for CXC, p. 198).
Measuring the Unit Length of a Coil
Measure the length of one loop of the coil by placing the tip of the tape measure at the end of the coil and wrapping it around the coil until it appears to reach the same horizontal position when the coil is viewed from directly above. This gives one “unit length” of the coil. When the turns of the coil are of equal length, the length of the coil as a whole can be split into these unit length segments.
Deriving the Length of a Coil
Count the number of loops in the coil, to find the number of unit lengths it contains. Multiply the number of loops by the unit length measured earlier, to find the total length of the coil. This is then equal to the length of the cylinder the coil was created from, so its surface area can be calculated in the same way; using the equation A=2πrl (Maui, Surface Area of a Helical Coil?, Eng-Tips Forums).
Although this calculation is acceptable for the majority of real world applications, it is not accurately down to fractions of a millimetre. While in principle a coil is simply a cylinder that is coiled up, the process of coiling can stretch the outer edge of each loop while compressing the inner edge. This results in microscopic deviations in the length of each loop, which can alter the length slightly.
An alternate way of measuring the length of the coil is also useful if the coil contains loops of uneven sizes. Instead of stopping when the length of one loop has been measured, continue winding the tape measure down the whole coil. The length measured is then the length of the cylinder the coil was formed from. The surface area is then 2π times the radius (or π times the diameter) times the length, as before.
References
Furlonge, Errol, Mathematics for CXC, Nelson, 2000, accessed 01 July 2010.
Maui, "Surface Area of a Helical Coil?", Eng-Tips Forums, 2004, accessed 01 July 2010.
"Surface area formulas", Math.com, 2005, accessed 01 July 2010.
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