Math: What is a Radian?

When you tutor math, you explain radians every semester to your grade 12 students.

Most people begin measuring angles in degrees.  However, you can also measure an angle in radians.  1 rad≈57.3°.

While degrees come from (I am told) Babylonia, or one of the ancient civilizations of that area, radians are a “natural” way to measure angles.  Behold:


In the above picture, CA is a radius. The arc from A to B is the same length as CA. Therefore, angle ACB is 1 radian. 1 radian is the angle that you traverse by following an arc the length of the radius. Said another way, it’s the angle subtended by an arc one radius long.

Recall that the circumference of the circle is 2πr, where r is the radius.  Since 2πr is the exact circumference, 2π radians is exactly 360°.

Radians can be referred to as rads, but are usually stated without any unit. That’s how you can tell which way the angle is measured:  if it’s in degrees, it will have a degree sign.  If it’s in rads, it won’t have any units.  Therefore, an angle of 54° means, of course, 54 degrees.  However, an angle of 32 means 32 rads.

Please keep enjoying this fine summer!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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