Calculations at an archimedean or arithmetic spiral. This is the simplest form of spirals, where the radius increases proportionally with the angle. The radius is the distance from the center to the end of the spiral. Enter radius and number of turnings or angle. Choose the number of decimal places, then click Calculate. Please enter angles in degrees, here you can convert angle units. The surface area of the archimedean spiral can here only be calculated if the number of turnings is an integer.
Formulas:
n = φ / 360°
r = a * φ (φ as radiant)
r1 = a * 360° = r / n
l = a / 2 * [ φ * √ 1 + φ² + ln( φ + √ 1 + φ² ) ]
p = ln(φ) + r for n≤1
p = ln(φ) - ln(φ-360°) + r1 for n>1
d = a * φ + a * (φ-180°) for φ>180°
κ = ( φ² + 2) / [ a * ( φ² + 1)3/2 ]
n
A = 4/3 π³ a² +
Σ
8 ( i - 1 ) π³ a²
i=2
Σ is the sum symbol, ln is the logarithmus naturalis (natural logarithm).
pi:
π = 3.141592653589793...
Radius, parameter a, length, perimeter and diameter have a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter). The number of turnings is dimensionless. The unit of the curvature is 1 / length unit.
The Archimedean spiral was discovered by Archimedes' friend Conon of Samos and was first described mathematically by Archimedes, at least as far as is known. This description was made in 225 BC in his work "On Spirals". It is called an arithmetic spiral because arithmetic means something like countable and the radius of such a spiral increases evenly with each rotation. This is in contrast to another type of spiral, the logarithmic spiral. An Archimedean spiral is created, for example, when a ribbon of constant thickness is wound up. Archimedean spirals and shapes based on them are a common element in design and as decoration in architecture. An Archimedean spiral always fits into a circle with the same radius, whereas it does not completely fill it.