What is a Angle Unit Converter?
The angle unit converter switches between degrees (°), radians (rad), and gradians (grad).
How it's calculated
180° = π radians, 1 rad ≈ 57.2958°, 360° = 400 grad
Units that are easy to mix up
| Units | Why it's confusing | How to tell them apart |
|---|---|---|
| degrees vs radians | Both measure angle | 180° = π rad; everyday use favors degrees, math/physics favors radians |
| gradians vs degrees | Both from French decimal attempts | 360° = 400 gradians; gradians never gained wide adoption |
Why radians exist
Degrees are an arbitrary human-chosen number, but radians derive directly from the circle itself. One radian is the angle subtended by an arc equal in length to the radius, so arc length = radius × angle (in radians) — a beautifully simple relationship. That simplicity is exactly why calculus (the derivatives of sine and cosine) only comes out clean when angles are in radians. Radians are the unit mathematics itself prefers.
Good to know
- The 360-degree circle traces back to Babylon — they measured a year at roughly 360 days, and their base-60 counting system made 360 (with 24 divisors) especially convenient to work with.
- The gradian (gon) was created alongside the metric system during the French Revolution, dividing a right angle into 100 parts — but unlike the meter, it never caught on widely.
- Slope is expressed three different ways — degrees, percent, and ratio. A road sign reading '10% grade' means 10m of rise per 100m of horizontal distance, not 10 degrees — that actually works out to about 5.7 degrees.
- Minutes and seconds of angle use the same base-60 system as time, and latitude/longitude coordinates use them — one minute of latitude equals exactly one nautical mile.
- On a clock, the hour hand moves 30° per hour (0.5° per minute), while the minute hand moves 6° per minute.
Frequently asked questions
Q. How many degrees is 1 radian?
About 57.2958°. Calculate as 180 ÷ π.
Q. What angle is a 10% grade?
About 5.71°. Calculated as arctan(0.1) — percent grade and degrees are not proportional.
Q. Why does calculus use radians?
Because the derivative formulas for sine and cosine only simplify correctly when angles are in radians; using degrees introduces a constant π/180 factor everywhere.