Understanding angle
One turn, three habits.
Degrees for humans. Radians for mathematics. Turns for the honest truth.
The full turn.
A complete circle is one turn. Everything else is a way of subdividing it. Degrees split it into 360 — a number inherited from Babylonian astronomy because 360 has many divisors and roughly matches the days in a year. Radians split it into 2π — the natural choice for calculus, because the derivative of sin(x) is just cos(x) when x is in radians.
1 turn ≡ 360° ≡ 2π rad ≡ 400 grad
Practical equivalences.
The numbers worth knowing
- π rad = 180°
- π/2 rad = 90° (right angle)
- π/3 rad = 60° · π/4 rad = 45° · π/6 rad = 30°
- 1° = 60 arcminutes = 3600 arcseconds
- 1 grad = 0.9°
The gradian (or gon) is metric's attempt at angle: 100 grad in a right angle, 400 in a full turn. It survives in surveying and on some calculator keyboards, but never quite caught on.
Three worked conversions.
90 degrees to radians
180° = π rad
Multiply by π and divide by 180.
90 × π ÷ 180 ≈ 1.5708
= ≈ π/2 rad
1 radian to degrees
π rad = 180°
Divide by π and multiply by 180.
1 × 180 ÷ π ≈ 57.2958
= 57.2958°
30 arcminutes to degrees
1° = 60 arcminutes
Divide — arcminutes are smaller than degrees.
30 ÷ 60 = 0.5
= 0.5°
A note on radians.
Strictly, the radian is dimensionless — it's the ratio of arc length to radius, two lengths cancelling out. That's why you can take the sine of a "number" without specifying units when the number is in radians. In every other angle unit, the result of a trig function comes out wrong unless you convert first.
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