Angle Conversions — Degrees, Radians and Gradians
Every scientific calculator has a DEG/RAD/GRAD mode switch, and it exists because angles genuinely get measured in three different systems. Degrees (°) are what you learn geometry in. Radians are what calculus, physics (angular velocity, simple harmonic motion) and every "advanced" formula actually use. Gradians (also called gons) are rarer today but still appear in surveying and some European engineering contexts, and in the "GRAD" button on your calculator that everyone eventually presses by accident. Getting the wrong mode selected is one of the most common silent errors in trigonometry — the calculator gives a real number, it's just the wrong one, and nothing flags it as an error.
degrees = radians × (180 ÷ π)
gradians = degrees × (10 ÷ 9)
degrees = gradians × (9 ÷ 10)
Full circle: 360° = 2π rad = 400 gradians
Why three systems exist: degrees split a circle into 360 arbitrary parts (a convention going back to Babylonian base-60 mathematics). Radians define the angle by the arc length it cuts on a unit circle — 1 radian is the angle where the arc length equals the radius — which is why radians make calculus formulas (like d/dx[sin x] = cos x) work cleanly without extra conversion constants. Gradians split a right angle into exactly 100 parts (so a full circle is 400 gradians), which was designed for decimal-friendly surveying calculations.
rad = 45 × (π ÷ 180) = 45π/180 = π/4 = 0.7854 rad (4 d.p.).
Check: 45° is one-eighth of a full circle (360°). One-eighth of 2π radians is 2π/8 = π/4. Matches.
deg = 2.5 × (180 ÷ π) = 2.5 × 57.2958 = 143.24° (2 d.p.).
90° → 90 × (10/9) = 100 gon (a right angle is exactly 100 gradians — this is the whole point of the gradian system).
180° → 180 × (10/9) = 200 gon.
Check: 200 gon → 200 × (9/10) = 180°. Matches.
In DEG mode: sin(30°) = 0.5 exactly.
In RAD mode, the calculator instead computes sin(30 radians). Since 30 radians = 30 × (180/π) = 1718.87°, and reducing modulo 360° gives 1718.87 − 4×360 = 1718.87 − 1440 = 278.87°, sin(278.87°) ≈ −0.988.
Same keystrokes, completely different — and both numerically "valid" — answers. This is exactly why checking your calculator's mode before a trig calculation is not optional.
- Leaving the calculator in RAD mode while a question gives the angle in degrees (or vice versa) — the single most common cause of a "correct method, wrong final answer" trig question.
- Forgetting that π/4, π/3, π/6 etc. are radian values, not degree values that happen to use π — always convert explicitly if mixing formula-derived radians with a degree-based answer requirement.
- Assuming gradians are obsolete and never checked in an exam — some competitive-exam and surveying-adjacent numericals do specify gradians precisely to test whether you know the 400-gradian-per-circle convention.
- Rounding π too early (using 3.14 instead of a more precise value) in a multi-step radian calculation, which compounds the error.
| Angle | Degrees | Radians | Gradians |
|---|---|---|---|
| Right angle | 90° | π/2 ≈ 1.5708 | 100 gon |
| Straight angle | 180° | π ≈ 3.1416 | 200 gon |
| Full circle | 360° | 2π ≈ 6.2832 | 400 gon |
| Common exam angle | 60° | π/3 ≈ 1.0472 | 66.67 gon |
The calculator suite's unit converter has a dedicated Angle category for instant conversion between all three systems.
Open the Angle Converter →For structured practice with trigonometry and its role in physics and maths, ABC Chemistry runs Class 11–12 coaching through live online across India — see abcchemistry.in for details.