Steinmetz solid

The Steinmetz solid, a method for calculating the volume of a sphere, was first discovered and adopted by the ancient Chinese mathematician Liu Hui. The ratio of the volume of the Steinmetz solid to the volume of the inscribed sphere is . This model resembles a square box, hence the name Steinmetz solid.

When two cylinders of equal radii and perpendicular to each other are superimposed, their intersection forms a Steinmetz solid. This unique construction yields a square cross-section where the Steinmetz solid's intercepted portion intersects with the inscribed sphere's circle. The area ratio between these shapes is . Since the ratio of the area of each cross-section is , the Zu Geng principle (or Cavalieri's principle) implies that the final volume ratio is .