Torus Sector Calculator

Name: Torus Sector - Geometry Calculator
Author: Jürgen Kummer

Calculations at a torus sector, a piece cut out of a torus. The cut is made by two planes containing the axis of rotation. The size of the piece is determined by the angle of intersection originating at the center. An angle of 360° covers the whole torus.
Enter both radiuses of the original ring or horn torus and the angle of intersection. Choose the number of decimal places, then click Calculate. Please enter angles in degrees, here you can convert angle units.

Formulas:

L = \frac{4 \cdot π^{2} \cdot R \cdot r \cdot α}{360 °}

A = L + 2 \cdot π \cdot r^{2}

V = \frac{2 \cdot π^{2} \cdot R \cdot r^{2} \cdot α}{360 °}

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 is the curved part of the surface area.

The volume of a torus sector is calculated as the volume of the complete torus multiplied by the angle of the sector and divided by 360 degrees, or respectively 2π. The lateral surface area is calculated analogously. For the total surface area, the two equal-sized circular end faces are added to this lateral area. The calculation for a sector of a toroid, a toroid sector, is performed in the same way, except that the end faces added to the lateral surface are not circles but other two-dimensional shapes. The cross-section of a torus sector, like that of a toroid sector, is an annulus sector. The volume of a torus sector is equal to that of a right circular cylinder with the same diameter and the same arc length along the centerline.

A torus sector is mirror-symmetric about the plane that bisects the angle between the two bounding planes. Torus sectors occur approximately, for example, in pipelines and ducts where a change of direction is required, specifically with angles of 90 degrees for a left or right turn and 180 degrees, such as in a siphon.

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