Toroid Sector Calculator

Calculations at a toroid sector, a piece cut straight out of a toroid. The size of the piece is determined by the angle of intersection originating at the center. An angle of 360° covers the whole toroid.
Enter the angle of intersection as well as radius, base perimeter and base area of the original toroid. Choose the number of decimal places, then click Calculate. Please enter angles in degrees, here you can convert angle units.

	Angle of intersection (α):			Example: a toroid sector with an angle of intersection of 180°
	Radius (r):
	Base perimeter (p):
	Base area (B):
	Lateral surface (L):
	Surface area (A):
	Volume (V):
	Surface-to-volume ratio (A/V):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Radius and perimeter have the same unit (e.g. meter), the areas have 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 calculation of a toroid sector follows the same method as that of a torus sector. Its volume and lateral surface area are proportional to those of the full toroid, determined by the angle of intersection. Calculating the total surface area requires the additional inclusion of the two flat side faces. Depending on the configuration of these faces, they often present the greatest challenge in such calculations.
The volume of a toroid sector is equivalent to that of a general right cylinder or right prism sharing the same base area and a length equal to the arc of the centerline. Like a torus sector, a toroid sector is mirror-symmetrical about the plane that bisects the angle between the two bounding planes. In practice, toroid sectors are found in structures such as ventilation ducts and sections of galleries or tunnels. Such conduits often feature non-circular cross-sections, unlike a standard torus, adopting other shapes instead. In the case of a drivable tunnel, for instance, the cross-section often resembles an elongated semicircle to accommodate a flat roadway. A tunnel curve may follow the path of a circular arc and thereby form a toroid sector.

Last updated on 06/07/2026. Author: Jürgen Kummer

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