Gabriel Horn Volume and Area
Contrast a horn’s finite volume with its surface-area convergence thresholds.
Free theory articleContrast a horn’s finite volume with its surface-area convergence thresholds.
Concept & interpretation
Gabriel's Horn (also called Torricelli's trumpet) is a geometric figure that has infinite surface area but finite volume. This paradoxical property arises when revolving the curve y = 1/x (or more generally y = x^(-p)) around the x-axis for x ≥ 1.
Experiment
Use the Gabriel Horn Volume and Area module to investigate convergence properties for different powers of p.
- Set the Exponent p control to 1 (the classic Gabriel's Horn). Notice that the Infinite volume converges to π, while the Infinite lateral surface is marked as Divergent.
- Change the Cutoff R control from 4 to 16. Observe how the Truncated volume slowly approaches π, while the surface area bounds increase without bound.
- Now, set Exponent p to 2. Notice that both the volume and the lateral surface area are now convergent for the infinite horizon.
- Set Exponent p to 0.5. In this regime, neither the volume nor the surface area converges as R goes to infinity.