Cylindrical Arc Calculator

Name: Cylindrical Arc - Geometry Calculator
Author: Jürgen Kummer

Calculations for a cylindrical arc. When two cylindrical pipes of equal size meet at their ends at a right angle, they can be connected by a pipe bend. The closed version of a pipe bend is referred to here as a cylindrical arc. Two identical cylinders can be attached at the ends. This shape represents one-quarter of a torus and therefore a special case of a torus sector.
Enter the radius r of one of the base circles, as well as the bend radius a of the inner side of the cylindrical arc. a is also the radius of the central opening of the corresponding torus. Choose the number of decimal places, then click Calculate.

Formulas:

L = π^{2} \cdot (a + r) \cdot r

A = L + 2 \cdot π \cdot r^{2}

V = \frac{1}{2} \cdot π^{2} \cdot (a + r) \cdot r^{2}

pi:

π = 3.141592653589793...

The radiuses have the same unit (e.g. meter), the surface area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

The lateral surface area, which is the curved part of the surface, is one-quarter of the surface area of the corresponding torus. The volume is one-quarter of the torus's volume. The total surface area consists of the lateral surface area plus the areas of the two identical circles at the ends.
Such a cylindrical arc represents the interior of a hollow pipe elbow, defined by the opening radius r and the inner bend radius a. The outer surface of the pipe elbow has a different lateral area, with a larger value for r and a smaller one for a, corresponding to the pipe wall thickness. Water pipes, and fluid conduits in general, often feature such bends when the pipeline needs to change direction. In these cylindrical bends, the pipe's circular cross-section remains the same size as that of the straight cylindrical pipe sections connected to the bend. Consequently, the mean flow velocity remains constant for a given volumetric flow rate. This differs from a sharply bent cylinder, where the effective flow cross-section can decrease at the point of the kink. However, the curvature of a pipe elbow does give rise to additional pressure losses and secondary flows.

Calculation of volume and other dimensions: cylindrical arc

Last updated on 09/06/2026. Author: Jürgen Kummer

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