Angle Converter
Convert between Radian, Degree, Arcminute and more — step-by-step working shown.
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Common conversions from Degree
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What is an Angle Converter?
An angle converter helps you change angle measurements between different units like degrees, radians, and gradians. These units are used in various fields, from everyday geometry and navigation to advanced mathematics, physics, and engineering.
The most common units are degrees and radians. Degrees are familiar from school, where a full circle is 360 degrees. Radians are often used in higher-level maths and science because they simplify many formulas, especially in calculus. Gradians are less common but have specific uses in surveying.
How to Convert Degrees to Radians
Understand the Relationship
Know that 180 degrees is equal to π (pi) radians. This is the fundamental conversion factor you'll use.
Set Up the Conversion Factor
To convert degrees to radians, you need to multiply your degree value by a fraction that cancels out degrees and leaves radians. This fraction is (π radians / 180 degrees).
Perform the Calculation
Multiply your angle in degrees by (π/180). For example, to convert 90 degrees to radians: 90 × (π/180) = π/2 radians.
Conversions in context
Robot Arm Rotation
60 degrees → π/3 radians
Pendulum Swing
π/12 radians → 15 degrees
Surveyor's Measurement
72 degrees → 80 gradians
Old Compass Bearing
150 gradians → 135 degrees
Complex Function Angle
3π/2 radians → 300 gradians
Graphics Program Rotation
25 gradians → π/8 radians
Common mistakes to avoid
- Thinking π is always 3.14159 when converting angles:While π is approximately 3.14159, in angle conversions (especially to radians), π often represents 180 degrees. Using the numerical value too early can lead to less precise answers or answers not expressed in terms of π. Remember that π radians is exactly 180 degrees. When converting, keep π as a symbol in your answer unless a decimal approximation is specifically required.
- Multiplying by 180/π when converting degrees to radians:This is the inverse operation. Multiplying by 180/π would convert radians to degrees, not degrees to radians. It's a common mix-up of the conversion factor. To convert degrees to radians, multiply by (π/180). To convert radians to degrees, multiply by (180/π). Think about which unit you want to cancel out.
- Assuming 1 gradian is the same as 1 degree or 1 radian:These are three distinct units of angle measurement with different scales. Treating them as interchangeable will result in incorrect calculations. Know the relationships: 1 full circle = 360 degrees = 2π radians = 400 gradians. This means 1 gradian = 0.9 degrees, and 1 radian ≈ 57.2958 degrees.
Questions people ask
What is a radian and why is it used in maths?
A radian is a unit of angle measurement where one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. Radians are widely used in higher mathematics and physics because they simplify many formulas, especially those involving calculus, trigonometry, and oscillations. For example, the derivative of sin(x) is cos(x) only when x is in radians.
How do I convert degrees to radians?
To convert an angle from degrees to radians, you multiply the degree value by the conversion factor (π/180). This is because 180 degrees is equivalent to π radians. For instance, 45 degrees becomes 45 * (π/180) = π/4 radians.
When should I use radians instead of degrees?
You should use radians when working with advanced mathematical concepts like calculus, differential equations, or complex numbers, as formulas are often simpler and more natural in radians. Radians are also preferred in physics for rotational motion, wave mechanics, and engineering applications where angles relate directly to arc length or circumference. Degrees are generally used for everyday geometry, navigation, and when angles are given in a more intuitive, whole-number format.
What is a gradian and who uses it?
A gradian (also known as a gon or grade) is a unit of angle where a full circle is divided into 400 gradians. This means a right angle is 100 gradians. Gradians are primarily used in surveying and some European countries for land measurement and cartography, as they simplify calculations involving right angles and percentages of a full circle. They are rarely encountered in general mathematics or physics.
How many degrees is pi divided by 4 radians?
To convert π/4 radians to degrees, you multiply by the conversion factor (180/π). So, (π/4) * (180/π) = 180/4 = 45 degrees. Therefore, π/4 radians is equal to 45 degrees.