Why do calculators have DEG/RAD/GRAD modes?

Each mode changes how trig functions (sin, cos, tan) interpret their input. In DEG mode, sin(90) = 1. In RAD mode, sin(π/2) = 1. In GRAD mode, sin(100) = 1. Always check which mode your calculator is in before computing trig values — using the wrong mode gives a numerically plausible but completely wrong answer.

How many radians is 45 degrees?

45 × π/180 = π/4 ≈ 0.7854 radians. This is one of the most common reference angles in trigonometry (sin 45° = cos 45° = √2/2 ≈ 0.7071).

Why do math functions in programming use radians?

Radians make calculus formulas cleaner. The derivative of sin(x) = cos(x) only holds when x is in radians. In degrees, the derivative would be (π/180)cos(x), requiring a correction factor in every calculation. Since programming math libraries are designed around calculus and signal processing, radians are the natural choice.

Degrees vs Radians: When to Use Each and Why

Both degrees and radians measure rotation, but they serve different audiences. Radians are the mathematical and scientific standard — built into the SI system and every programming math library. Degrees are the everyday unit used in navigation, geography, and practical engineering. Engineers and programmers encounter both regularly and need to convert fluently between them.

Published March 20, 2026

Key takeaways

Radians are the SI unit for angle. 1 radian is the angle subtended when arc length equals the radius.
Full circle: 360° = 2π rad ≈ 6.2832 rad. Right angle: 90° = π/2 rad.
To convert degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
Radians are used in mathematics, physics, and all programming math libraries. Degrees are used in navigation, geography, and everyday contexts.

What is a radian?

A radian is defined geometrically: take a circle of any radius r, and lay an arc of length r along the circumference. The angle at the center subtended by that arc is exactly 1 radian. Because the circumference of a circle is 2πr, a full circle subtends 2π radians — approximately 6.2832 rad.

This definition makes radians 'natural' in the sense that they arise directly from the geometry of circles without any arbitrary scaling factor. When you work in radians, formulas in calculus, trigonometry, and physics simplify considerably. The arc length formula becomes s = rθ (not s = rθ × π/180). The derivative of sin(x) is simply cos(x) — no correction factor required.

What are degrees?

Degrees divide a full circle into 360 equal parts. The convention dates to Babylonian mathematics, which used a base-60 (sexagesimal) number system. The Babylonians divided the sky into 360 parts (roughly the number of days in a year), and this system was so convenient — 360 has 24 divisors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 — that it persisted through Greek and Arab astronomy into modern use.

Degrees can be subdivided into arc minutes (1/60 of a degree, symbol ′) and arc seconds (1/3600 of a degree, symbol ″). This system is still used in GPS coordinates (latitude/longitude), astronomy, and celestial navigation.

The conversion formula

Formula

degrees × (π / 180) = radians
radians × (180 / π) = degrees

Common reference angles:
0° = 0 rad
30° = π/6 ≈ 0.5236 rad
45° = π/4 ≈ 0.7854 rad
60° = π/3 ≈ 1.0472 rad
90° = π/2 ≈ 1.5708 rad
180° = π ≈ 3.1416 rad
270° = 3π/2 ≈ 4.7124 rad
360° = 2π ≈ 6.2832 rad

Memory shortcut: π radians = 180°. Multiply or divide by π/180 to convert.

When radians matter in practice

All trigonometric functions in standard programming math libraries expect input in radians. In Python, JavaScript, C, Java, and virtually every other language, Math.sin(), Math.cos(), and Math.tan() take radians. If you pass degrees without converting first:

Python: import math; math.sin(90) returns approximately −0.8940 — not 1.0 — because 90 is interpreted as 90 radians (about 28.6 full rotations).

JavaScript: Math.sin(90) similarly returns −0.8940 for the same reason.

The fix: always convert first. math.sin(90 * math.pi / 180) = math.sin(π/2) = 1.0 exactly.

In physics and calculus, radians are essential for clean formulas. Angular velocity ω is in rad/s. The derivative d/dx(sin x) = cos x holds only when x is in radians; in degrees it would be (π/180)cos x. Fourier analysis, wave mechanics, and complex exponentials (Euler's formula e^(iθ)) all use radians natively.

In engineering and navigation, degrees dominate. CAD software typically works in degrees. Surveying bearings are in degrees, minutes, seconds. Machine tool angles, CNC programs, and most industrial controllers use degrees. GPS coordinates use decimal degrees or degrees-minutes-seconds.

Gradians — the third system

Note

A third angle unit exists: the gradian (also called gon or grad). A full circle = 400 gradians, and a right angle = 100 gradians. This was designed to make right-angle arithmetic cleaner (100 grad = 90°). Gradians are used in surveying in some European countries, particularly France, Switzerland, and the Netherlands. Scientific calculators include a GRAD mode alongside DEG and RAD. Outside surveying in those specific regions, gradians are rare — you are unlikely to encounter them unless working with European survey data.