Angle Converter
Convert between degrees, radians and gradians — and see whether your angle is acute, right, obtuse or reflex.
ANGLE CONVERTER
Main units
Precise (astronomy & navigation)
Enter an angle
Type a value above to classify the angle
Common angles — click to load
Step-by-step solution
Type an angle to see the conversion working
The conversion formulas
π = 3.14159265… · 1 radian = 180/π ≈ 57.296°
Key angle facts
Real-world angles
What is a radian? 🔵
One radian is the angle at the centre of a circle where the arc length equals the radius. Because a full circle's circumference is 2πr, there are exactly 2π radians in 360°.
Angle Converter Explained
Core concepts, classroom applications, and plain-English explanations for all audiences.
Where you'll actually use this
A clock's hour hand moves 360° in 12 hours — that's exactly 30° per hour, or 0.5° per minute. Sailors and pilots navigate using compass bearings in degrees (0–360°). Builders quote roof pitch in degrees. But when you write a rotation animation in JavaScript or Python, the maths library expects radians, not degrees. Knowing how to convert is a genuine skill gap between beginners and professionals.
Worked example
"Your code needs to rotate an element by 45°. The function takes radians. What do you pass in?"
Formula: radians = degrees × π ÷ 180
45 × π ÷ 180 = 45 × 0.01745...
= 0.7854 rad
Result: Pass in π/4 (or 0.7854) — not 45
Memory tip: 180° = π radians. Half a circle. Once you know that, all other conversions follow.
Key angle facts
- → 2π rad = 360°: A full circle. π rad = 180° (straight line). π/2 rad = 90° (right angle).
- → Gradians (°g): A right angle = 100 gradians. Used in some surveying equipment. A full circle = 400 gradians. Rarely seen outside civil engineering.
Why radians exist
Radians are the "natural" unit for angles. When you differentiate sin(x), you get cos(x) — but only if x is in radians. In degrees, you'd get an extra factor of π/180 everywhere. Calculus and physics both insist on radians.
Understanding Angles and Their Units
Angles are fundamental in mathematics, measuring the amount of rotation between two lines or surfaces. From geometry to trigonometry, understanding how to measure and convert angles is crucial. Our Angle Converter helps you seamlessly switch between various units, making complex calculations simpler and ensuring accuracy in your work.
This tool supports conversions between degrees, radians, gradians (gon), turns, arc minutes, and arc seconds. For GCSE and A-level Maths, the primary focus is often on degrees and radians. You'll find key conversions like 30° = π/6 radians, 45° = π/4 radians, 60° = π/3 radians, 90° = π/2 radians, 180° = π radians, and 360° = 2π radians, alongside classifications such as acute, right, obtuse, straight, reflex, and full turn angles.
A core concept is the definition of a radian – the angle subtended at the centre of a circle by an arc equal in length to the radius. The essential formula for converting degrees to radians is to multiply the degree value by π/180. This page provides the context and explanations you need to master angle conversions for your studies.
Common mistakes to avoid
- To convert 90 degrees to radians, I divide by 180:You've forgotten to include π in the conversion factor. Dividing by 180 alone won't give you the correct radian value. To convert degrees to radians, multiply by π/180. So, 90° × (π/180) = π/2 radians.
- I thought 100 gradians was the same as 100 degrees:Gradians (gon) divide a full circle into 400 units, while degrees divide it into 360 units. They are different scales, so 100 gradians is not equal to 100 degrees. 100 gradians is equivalent to 90 degrees (a right angle), as 400 gradians = 360 degrees.
- When calculating arc length, I used the formula s = rθ with θ in degrees:Formulas for arc length (s = rθ) and area of a sector (A = ½r²θ) are derived assuming θ is in radians. Using degrees will give an incorrect result. Always convert your angle to radians before using formulas like s = rθ or A = ½r²θ. For example, if θ = 60°, convert it to π/3 radians first.
- I converted 120 degrees to 120π/180 radians and left it like that:While technically correct, it's not in its simplest form. You should always simplify fractions in mathematics to make them easier to work with and understand. Simplify the fraction: 120π/180 = (120 ÷ 60)π / (180 ÷ 60) = 2π/3 radians. Always reduce fractions to their lowest terms.