Angle Converter — Degrees, Radians & Gradians

Angle Conversion Guide

What do I need to know about 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 Standard reference values for trigonometric functions and geometry are also available in <a href="https://247tables.com" rel="noopener">247 Tables</a>. For other measurement conversions, <a href="https://247quicktools.com" rel="noopener">247 Quick Tools</a> handles length, weight, and temperature.

What's the key thing to understand about 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 gons, divide a full circle into 400 units instead of 360 degrees or 2π radians, and are occasionally used in surveying and some European engineering contexts.

What's the key thing to understand about 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.