Angles and How We Measure Them

There is nothing geometrically special about the number 360 — so mathematics uses a more natural unit, where an angle is measured by the arc it sweeps. This lesson builds a precise language for angles. We start with degrees and their sexagesimal subdivisions — arcminutes and arcseconds — and convert between degrees-minutes-seconds and decimal degrees. Then we introduce the radian, defined so that an arc of length equal to the radius subtends exactly one radian, which makes the arc-length formula s = rθ fall out with no conversion factor at all. We finish with the conversions every trig problem relies on and the standard angle equivalences worth memorizing. In this lesson: Degrees, arcminutes, and arcseconds (1° = 60′ = 3,600″) Converting between DMS and decimal degrees (worked example) The radian: arc length equal to the radius, and 2π for a full turn Arc length s = rθ and sector area A = ½r²θ Converting between degrees and radians (×π/180 and ×180/π) The standard degree–radian equivalences This is Section 2 of Trigonometry, an 11-section chapter from The READY Academy. The full chapter playlist is linked in the description. #trigonometry #radians #mathematics #stem 00:00 Intro 00:21 Degrees, Minutes, and Seconds 04:55 Radians: the Natural Unit 09:21 Converting Between Degrees and Radians