For centuries, the Chinese had simply approximated the value of pi as 3. However, Liu Xin approximated it to 3.154 between 1–5 CE by use of an unknown method. Standard measuring vessels dating to the reign of Wang Mang from 9–23 CE also showed approximations for pi at 3.1590, 3.1497, and 3.167. Zhang Heng was the next known Han mathematician to have made an approximation for pi. Han mathematicians realized that the area of a square versus the area of its inscribed circle had a ratio of 4:3, and also understood that the volume of a cube and the volume of its inscribed sphere would thus be 42:32, or 16:9. With D as diameter and V as volume, D3:V = 16:9 or V=9⁄16D3, a formula Zhang found fault with since he realized the value for diameter was inaccurate, the discrepancy being the value taken for the ratio. To account for the inconsistency, Zhang added 1⁄16D3 to the formula, making V = 9⁄16D3 + 1⁄16D3 = 5⁄8D3. Since he found the ratio of the volume of the cube to the inscribed sphere at 8:5, the ratio of the area of a square to the inscribed circle is √8:√5. With this formula, Zhang was able to approximate pi as the square root of 10, or 3.162.