Investigation 6.6.2 - Gradian Measure

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Investigation 6.6.2 - Gradian measure Have you ever wondered why we have 360 degrees in a circle when our number system is base 10? Why not divide the circle into 10 angular units? Or 100? The notion of the “degree” as the base angular unit of measure is generally attributed to the Babylonians. The Babylonian numerical system was sexagesimal (base 60). The number “360” played an essential role in their calendar as it is close to the number of days in a year. Another hypothesis is that the circle divides more naturally into six parts than it does four or ten. Try drawing diameters through a circle cutting it into even parts. I. 6 equal parts II. 4 equal parts III. 10 equal parts Then draw chords by connecting consecutive diameter endpoints. Which method produces equilateral triangles? Which dissected circle has chords congruent to its radius? Now find the central angle of each dissection. Which is most closely related to the Babylonian sexagesimal number system? Now suppose we wanted to derive an angular unit of measure more closely related to our base 10 decimal system. Let us begin by dividing a right angle into 100 units, called “gradians”. (You may have this system on your calculator, denoted “grad”.) a. How many gradians are in one whole circle? b. Convert 180° into gradians. c. How many degrees make 50 gradians? If societies in science fiction have differing number of days in their respective years due to different planetary orbits, it would stand to reason that their methods for measuring angles and circles might be different from our system. But would the formulas for calculating the circumference and area of a circle also be different? Or does a circle with radius r = 1 still have a circumference of 2π no matter where it is in the universe?.

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