Annulus Stripe Calculator

Name: Annulus Stripe - Geometry Calculator
Author: Jürgen Kummer

Calculations with an annulus stripe. If an annulus is divided by two parallel lines running within the inner circle in equal distance from the center, two identical annulus stripes are created. One of these two can be calculated here; the upper annulus stripe is shown in the sketches.
Enter the radii of the two circles and the stripe spacing. The stripe spacing is the distance of one of the two parallel lines from the center of the circles; it must not be greater than the inner radius. Choose the number of decimal places and click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units.

Sketch annulus stripe with θ, φ, a and h

Construction of an annulus stripe from an annulus ring and two parallel lines.

Formulas:

θ = \arcsin (\frac{a}{R})

φ = \arcsin (\frac{a}{r})

h = \sqrt{R^{2} - a^{2}} - \sqrt{r^{2} - a^{2}}

p = R \cdot θ + r \cdot φ + 2 h

A = a \cdot \sqrt{R^{2} - a^{2}} + \frac{R^{2} \cdot θ}{2} - a \cdot \sqrt{r^{2} - a^{2}} + \frac{r^{2} \cdot φ}{2}

Radii, spacing, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The annulus stripe is bounded at the top and bottom by circular segments. Other shapes cut out of an annulus are the annulus sector and the annulus segment. One shape that is somewhat likely to be confused with the annulus stripe is the curved rectangle. In that case, the underlying circles are the same size and not concentric.
For an annulus stripe where the two bounding lines are not the same distance from the center, the area can be calculated as follows: First, calculate the distance of the left line and halve the calculated area value, then calculate the distance of the right line; this area is also halved. Now, these two values are added if the lines are opposite each other and subtracted if they are on the same side, both as seen from the center. The perimeter of such a shape is determined in a similar way; you just have to be careful to include the side lines correctly.

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