Angle Converter — Degrees, Radians & Gradians

Angle Conversion Guide

Conversion Factors

Full circle: 360° = 2π radians = 400 gradians = 1 turn. Conversion: degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Degrees to gradians: multiply by 10/9. Gradians to degrees: multiply by 9/10. Key radian values to know: 30° = π/6 rad. 45° = π/4 rad. 60° = π/3 rad. 90° = π/2 rad. 180° = π rad. 270° = 3π/2 rad. 360° = 2π rad. Radians are the natural unit for calculus — all standard differentiation and integration rules for trig functions (d/dx sin(x) = cos(x)) only w

Arc Length and Sector Area

Arc length: s = rθ (where θ is in radians, r is radius). Arc length of a 60° sector with radius 5: θ = π/3 rad. s = 5 × π/3 = 5.24. Sector area: A = ½r²θ (θ in radians). 60° sector, radius 5: A = ½ × 25 × π/3 = 13.09 square units. These formulas are cleaner in radians than degrees — one reason radians are preferred in mathematics and physics. If θ is in degrees: arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². The radian form is simpler because 2π/360 factors cancel cleanly.

Why Radians Exist

The radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. Because of this definition, arc length = radius × angle (in radians) — a beautifully simple relationship. In calculus: the derivative of sin(x) is cos(x) ONLY when x is in radians. In degrees: d/dx sin(x°) = (π/180) cos(x°). In physics and engineering, angles in dynamic equations (angular velocity ω = v/r, angular acceleration α) are always in radians per second. Gradians (also called

Trigonometric Values

Key trig values for exact answers: sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1. cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 0.5, cos(90°) = 0. tan(0°) = 0, tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3. These are essential for non-calculator paper work in GCSE and A-level maths. The values can be derived from equilateral and right-isosceles triangles — drawing these triangles and applying SOHCAHTOA gives exact values without memorisation.