The most commonly seen Archimedean spirals are that of tightly wound springs, edges on rolled up rugs, and decorative pieces on jewelry. More practical uses are used in machines such as the sewing machine. The machine transforms rotary motion to linear motion. The Archimedean spring is particularity known for being able to respond to both torsional and translational force.
The spiral has the property that any ray from the origin intersects multiple points on the spiral with constant distance between each point. That distance is said to be 2pib if it is measured in radians. The idea is that the spiral has two arms one when theta is positive and one when theta is negative. These arms connect at the origin or center of the spiral. Only one however is shown, the other is not and if was would look like the mirror image of the showing arm across the y-axis. Below I have listed a link that will take you to a interactive graph that allows you to adjust a and b to see how the formula works.
We know that because there is a linear relation between the radius and the angle, the distance between each coil will be constant. Therefore, for this to be true, we know that the arms get bigger by a constant rate as well with each turn. To find the entire length of the spiral, we need to use the equation r = a+bx, where r is the distance from the origin, x represents theta. a represents the starting point of which the spiral rotates and b represents the distance between each spiral. The total angle turned in the spiral is 2pib, therefore the distance between each loop is 2pib divided by the number of turns.