Gabriel’s Horn: The Volume Paradox

Imagine a funnel that never ends yet holds only a finite amount of liquid. That’s Gabriel’s Horn—also called Torricelli’s Trumpet—and it flips our intuition about size on its head. By rotating the curve y=1/x around the x-axis from x=1 to infinity, you get a shape with a surprisingly small interior but a bafflingly vast surface area. It’s a paradox that uses simple algebra, but leads straight into the heart of calculus and geometry.

Where did this come from?

In the 1640s, Italian mathematician Evangelista Torricelli (a disciple of Galileo) studied the curve y=1/x and its ‘‘trumpet’’ shape. He found that while its volume converges to π cubic units, its surface area diverges to infinity. The name Gabriel’s Horn was coined later—an allusion to the angel Gabriel’s trumpet in the Bible, blowing endlessly. Torricelli’s discovery was one of the earliest glimpses of what we now call ‘‘integral calculus.’”},{