Archimedes used two approximation methods of circumscribing regular polygons and inscribing regular polygons to study pi.

This is actually the case. He first used the method of infinite division to prove that the area of ??a circle is equal to a right triangle, whose right-angled side is the radius of the circle, and whose right-angled side is the circumference of the circle. Liu Hui's method is the same. He first proves that "the half-circumference radius multiplied by the product step" means that half of the perimeter is multiplied by the radius to obtain the area of ??the circle.

The logic involves area and length. It will be more intuitive if you use the method of calculating area. Obviously, a circle is smaller than its circumscribed polygon and larger than its inscribed polygon. Using length is equivalent to using area calculations. Archimedes calculated lengths, or more precisely ratios of lengths, that is, proportions.

What Liu Hui calculated was the area:

This equivalent principle can be seen from Liu Hui’s method of calculating the area: When Liu Hui calculated the area of ??the 2N polygon, Directly use the side length of the N polygon.

Archimedes gave a surprising conclusion at the beginning, because it was given directly without any derivation. And as his student, it is easy to prove this. The Greeks liked proportions and fractions, and decimals may not have been popular at that time. So, how accurate is this score? The 4 digits after the decimal point are all the same, and the 5th digit is only slightly larger.

So I say, Archimedes is amazing. Archimedes doesn't tell you how such a score is derived. Because only if you study hard, you will cherish what you get. "If it's easy to get, just wait and see." This is how Archimedes should educate his students, so he doesn’t say how to get it.