DEG vs RAD vs GRAD — Which Mode and When
Every scientific calculator has at least two angle modes: DEG and RAD. Many also include GRAD. The mode tells the calculator how to interpret an angle when you press sin, cos, or tan. Pick the wrong one and the answer is wrong by an order of magnitude.
Degrees (DEG)
Degrees split a full circle into 360 equal parts. A right angle is 90°. A straight line is 180°. School-level trigonometry, navigation, and most everyday angle work uses degrees. The notation 30°, 45°, 60° is the degree mark. On the calculator, DEG mode treats your typed angle as degrees — so sin 30 in DEG mode gives 0.5.
Radians (RAD)
Radians measure angles in terms of the arc length on a unit circle. A full circle is 2π radians (≈ 6.283). A right angle is π/2 rad (≈ 1.571). Radians are the natural unit for calculus — the derivative of sin(x) equals cos(x) only when x is in radians. Physics, engineering, and most maths past A-Level use radians by default. In RAD mode, sin(π/6) = 0.5.
Gradians (GRAD)
Gradians (also called gons) split a full circle into 400 parts. A right angle is 100 grad. Used in surveying, civil engineering, and some artillery applications. Rare outside those fields. Most students will never need GRAD mode — but knowing it exists prevents confusion when the indicator shows GRAD by accident.
Conversion formulas
Between degrees and radians: radians = degrees × π/180, degrees = radians × 180/π. So 90° = π/2 rad ≈ 1.5708; 1 rad ≈ 57.296°. Between degrees and gradians: gradians = degrees × 10/9. So 90° = 100 grad. Between radians and gradians: gradians = radians × 200/π. The conversions are exact; the decimal approximations come from rounding π.
Which mode to use when
School trigonometry up to A-Level / Class 12: DEG. Pure calculus and most university maths: RAD. Engineering depends on the textbook — many use degrees for design specs and radians for analysis. Physics: RAD for theory, DEG for measurements. Surveying: GRAD. When in doubt, look at how the problem states angles — if it says '30°', use DEG; if it says 'π/6' or just '0.5236', use RAD.
Spotting a wrong-mode answer
If sin of an angle returns a value you did not expect, check the mode. sin 30 should be 0.5 in DEG mode, but −0.988 in RAD mode. tan 45 should be 1 in DEG mode, but 1.620 in RAD mode. The numbers giveaway are extreme — if you got something close to but not exactly 0.5 or 1 for a standard angle, the mode is wrong. Switch and recompute.
Frequently asked questions
Why are there three angle modes?
Different fields use different conventions. Degrees are familiar from school and everyday speech. Radians are mathematically natural — calculus formulas only work cleanly in radians. Gradians simplify decimal-based engineering work (a right angle is 100 grad, exactly). Calculators include all three to serve every field.
Which mode should I use for school work?
Degrees. UK GCSE, A-Level, Indian boards, and US high-school maths all use degrees for trigonometry. The exception is calculus topics in A-Level or Class 12 maths, which switch to radians. Always check the question — if it shows the degree symbol or whole numbers like 30, 45, 60, use DEG.
What does the mode indicator look like?
Most calculators show a small DEG, RAD, or GRAD label in the corner of the display. Online calculators may use a coloured pill or button. The indicator never goes away — it tells you the current mode at all times. Check it before every trig calculation.
How do I convert 90 degrees to radians?
Multiply by π/180. So 90° = 90 × π/180 = π/2 ≈ 1.5708 radians. Other common conversions: 30° = π/6, 45° = π/4, 60° = π/3, 180° = π, 360° = 2π. Memorising the first six is enough for most exam work.
Why does calculus use radians?
The derivative of sin(x) is cos(x) — but only if x is in radians. If x is in degrees, you get a constant factor of π/180 every time you differentiate. Radians make the constant disappear, which is why every calculus identity and Taylor series uses them. The convention is mathematical convenience, not arbitrary.