Angle Converter
Convert an angle between degrees and radians, and see the result expressed as a multiple of π.
How to use the angle converter
- Enter the angle you want to convert.
- Choose whether the angle is given in degrees or radians.
- The calculator shows the angle in radians, in degrees and as a multiple of π automatically.
How the angle is converted
Degrees and radians are two different ways of measuring angles. The relationship between them is:
degrees = radians × 180 ÷ π
Worked example: 90 degrees becomes 90 × π ÷ 180 = 1.5708 radians, which can also be written 0.500 × π. The relationship is easy to remember from the fact that a half circle is both 180 degrees and π radians — everything else follows from that ratio. Radians feel awkward in everyday use, but they are required in mathematics because results such as the derivative of sine only hold when the angle is measured in radians.
A full circle is 360°, which equals 2π radians. That's why many textbook answers are expressed as fractions of π — for example, 90° is the same as 0.5 × π radians, and 180° is the same as π radians.
Percent grade, per mille and degrees describe the same slope
Outside mathematics, slope is rarely given in degrees. Roads and railways use percent or per mille gradient — the rise divided by the horizontal run. A gradient of 10 % means one metre up for every ten metres forward, which corresponds to roughly 5.7 degrees: noticeably less than most people guess, because percent and degrees do not track each other linearly.
The relationship runs through the tangent: percent is the tangent of the angle multiplied by a hundred. That means the gap between the two scales widens as the slope gets steeper. At 45 degrees the gradient is 100 %, and a gradient of 200 % corresponds to about 63 degrees — the percent scale has no upper limit, while the degree scale stops at 90.
Frequently asked questions
Why do mathematicians often use radians instead of degrees?
Radians tie the angle directly to the length of the circular arc and radius, which makes formulas in calculus and trigonometry simpler and more consistent. Many formulas for differentiating and integrating trigonometric functions assume radians.
How many radians is a full circle?
A full circle is 360°, which equals 2π radians (about 6.2832 radians).
What is π (pi) in this context?
π is the ratio between a circle's circumference and diameter, about 3.14159. In angle conversion, π is used to link degrees (where a full circle is 360) with radians (where a full circle is 2π).