Circular Segment Revolution Volume

Find the volume of the solid formed by revolving a circular segment about an axis, using the Pappus theorem from radius, central angle, and axis distance.

Input

Compute the volume of the solid formed by revolving a circular segment about an axis, using the Pappus theorem. Enter the circle radius, central angle, and distance to the axis.

Circle radius r

Central angle θ (degrees)

Distance to axis d

Result

Volume V

1,111.894825

Segment area A

15.354621

Centroid to axis Rc

11.525101

Centroid to center

3.525101

Chord length c

8.660254

Sagitta h

2.5

This uses the Pappus theorem V = 2π × Rc × A, where Rc is the radius of the circle traced by the centroid.

Lengths use the same unit as the inputs, areas its square, and volume its cube.

How it works

The segment area is A = (1/2) r² (θ − sin θ), where r is the circle radius and θ is the central angle in radians.
The distance from the circle center to the segment centroid is yBar = (4 r sin³(θ/2)) / (3 (θ − sin θ)).
With d as the distance from the rotation axis to the circle center, the centroid-to-axis distance is Rc = d + yBar (the segment bulges away from the axis).
By the Pappus theorem, the revolution volume is V = 2π × Rc × A, the segment area times the circumference traced by its centroid.
The chord length is c = 2 r sin(θ/2) and the sagitta (segment height) is h = r (1 − cos(θ/2)).
Lengths use the same unit as the inputs, areas use its square, and volume uses its cube. The axis is assumed not to intersect the segment.

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