Angle Converter
Convert angles between degrees, radians, gradians, and other units used in mathematics, navigation, surveying, and astronomy. Enter any angle and see the equivalent in all common angular measurement systems at once.
How It Works
An angle measures the amount of rotation between two lines meeting at a point. This converter uses degrees (°) as its base unit, storing each unit's factor as the number of degrees it represents.
The degree (°) divides a full circle into 360 parts. This choice traces back to ancient Babylon, whose base-60 number system made 360 convenient (divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180). Degrees are further divided into arcminutes (1° = 60 arcminutes) and arcseconds (1 arcminute = 60 arcseconds), a system called DMS (degrees, minutes, seconds). Astronomers and land surveyors use DMS notation extensively.
The radian (rad) is the SI unit of angle and the natural unit for mathematics. It is defined as the angle subtended when the arc length equals the radius. A full circle = 2π radians ≈ 6.2832 rad. Since 360° = 2π rad, 1 rad = 180/π ≈ 57.2958°. Radians appear in all trigonometric calculus (derivatives of sin/cos are only clean in radians) and in physics formulas for angular velocity and frequency.
The gradian (grad or gon) divides a right angle into 100 parts, making a full circle 400 gradians. Introduced during the French Revolution as part of metrication, gradians are still used in surveying in some European countries, particularly France and Germany. A 90° right angle = 100 grad; a straight line = 200 grad.
The milliradian (mrad) is used in military range-finding and scope adjustments. At 1,000 meters distance, 1 mrad corresponds to 1 meter of adjustment — a convenient relationship that makes mrad natural for ballistics.
Worked example — Convert 45° to radians:
45° × π/180 = π/4 ≈ 0.7854 rad
Worked example — Convert π/3 radians to degrees:
π/3 × 180/π = 60°
Worked example — Surveying with gradians:
A bearing of 150 grad = 150 × 0.9 = 135° from North = Southeast direction.
Worked example — Telescope pointing:
The Moon subtends about 0.5° = 30 arcminutes = 1,800 arcseconds = 8.727 mrad in the sky.
Frequently Asked Questions
Why are there 360 degrees in a circle?
The Babylonians used a base-60 (sexagesimal) number system and divided the sky into 360 parts corresponding roughly to the days in a year. Because 360 has many factors (24 divisors), fractions of a circle come out as whole numbers in many practical situations.
What is a radian?
A radian is the angle at the center of a circle that subtends an arc equal in length to the radius. Since the circumference is 2πr, a full circle = 2π rad ≈ 6.2832 rad. It is the natural unit for angle in mathematics because it simplifies formulas in calculus and physics.
How many degrees is π radians?
π radians = 180°. This is the most important radian conversion to remember. π/2 rad = 90°, π/4 rad = 45°, 2π rad = 360° (a full circle).
What is a gradian (grad)?
A gradian divides a right angle into 100 equal parts, with a full circle being 400 grad. It was introduced during the French Revolution as part of the metric system. 90° = 100 grad, 180° = 200 grad, 360° = 400 grad.
What is an arcminute and arcsecond used for?
Arcminutes (1/60th of a degree) and arcseconds (1/3600th of a degree) are used in astronomy, GPS, and surveying to express very small angles. The Hubble Space Telescope can resolve details as small as 0.05 arcseconds. GPS coordinates typically give latitude/longitude to sub-arcsecond precision.