Semicircle Toroid Calculator

Calculations for two different semicircle toroids. Semicircle toroids are special toroids. They are created by dividing a ring torus along a circle with the same radius as the major radius of the torus. This circle is perpendicular to the plane of symmetry and runs from top to bottom through the torus. The result is a larger outer semicircle toroid and a smaller inner semicircle toroid with mirrored semicircles as cross sections. Both have the same cylindrical lateral surface where the torus was divided, which is curved in one direction. The surfaces curved in two directions are located on the outside for the outer semicircle toroid and on the inside for the inner semicircle toroid, opposite the cylindrical lateral surface.
Enter both radiuses and choose the number of decimal places, then click Calculate. The values for both different shapes are calculated.

	Major radius (R):			Outer and inner semicircle toroid
	Minor radius (r):
	Cylindrical lateral surface (L):
	Surface area outer toroid (Ao):
	Surface area inner toroid (Ai):
	Volume outer toroid (Vo):
	Volume inner toroid (Vi):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

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 formulas were derived using the theorems of Pappos. For the volumes, the second theorem of Pappos was used, according to which the volume of a solid of revolution equals the product of the area of the generating figure and the distance travelled by its centroid. The surface areas are obtained from the sum of the curved surface generated by the semicircular arc and the cylindrical lateral surface created by the cut. The curved surfaces were calculated using the first theorem of Pappos for surfaces of revolution. The theorems of Pappos are identical to Guldin's rules for solids of revolution. They were formulated by Pappos of Alexandria in the 4th century AD and rediscovered by Paul Guldin in the 17th century.

There are other semicircle toroids that are generated differently. If the ring torus is cut along the plane of symmetry, the result is half of a ring torus, whose calculation is trivial using the formulas for the torus and the annulus. However, the semicircles may also be inclined, in which case the calculation becomes considerably more complicated.

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

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