Why does circle have 360 degrees and not 100?

Answer: A circle is divided into $360^\circ$ rather than $100^\circ$ due to ancient astronomical calendars approximating a 360-day year, the Babylonian base-60 numeral system, and the superior divisibility of 360 (which has 24 factors, making angles like $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6}$ whole numbers, unlike 100).

Step-by-step solution

Step 1: Historical and Astronomical Origin

Ancient astronomers, particularly the Babylonians, tracked the Sun's apparent path across the sky against the background stars. They observed that the Sun takes roughly 360 days to complete a full circular orbit of the seasons, advancing by about $1$ degree each day.

Step 2: Sexagesimal Base-60 Number System

Babylonian mathematics used a base-60 (sexagesimal) numeral system. Just as 100 is significant in our base-10 system, 360 arose naturally as $6$ units of $60$ , which also matched six equilateral triangles fitting together around the center of a circle.

Step 3: High Divisibility of 360 vs 100

The number 360 is an exceptionally composite number with 24 integer divisors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. In contrast, 100 has only 9 divisors and cannot be divided evenly by 3, 6, 8, or 12, making fractions such as a third or a sixth of a circle awkward.

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