Angle Calculator: Complete Guide to Degrees, Radians, Angle Types, Trigonometry, and Angle Measurement
Angles are fundamental to geometry, trigonometry, physics, engineering, and virtually every branch of science. An angle measures the rotation between two rays sharing a common endpoint (the vertex). This calculator converts between all major angle units — degrees, radians, gradians, turns, and DMS (degrees-minutes-seconds) — classifies angle types, computes complementary and supplementary angles, displays trigonometric values, and provides an interactive protractor visualization. Whether you are a student, engineer, surveyor, programmer, or designer, understanding angle measurement and conversion is essential.
Degrees vs Radians
The two most common angle units are degrees (°) and radians (rad). A full circle is 360° or 2π radians. The conversion is: radians = degrees × π/180 and degrees = radians × 180/π. Key equivalents: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. Degrees are intuitive for everyday use (navigation, construction, weather), while radians are essential in calculus, physics, and engineering because they simplify formulas — the derivative of sin(x) is cos(x) only when x is in radians.
Radians → Degrees: ° = rad × 180/π
Degrees → Gradians: gon = ° / 0.9
Degrees → Turns: turns = ° / 360
30° = π/6 | 45° = π/4 | 60° = π/3
90° = π/2 | 180° = π | 360° = 2π
Complement: 90° − θ
Supplement: 180° − θ
Types of Angles
Angles are classified by their measure: Acute (0° < θ < 90°) — sharp, like the corner of a yield sign. Right (θ = 90°) — perpendicular, forming an L shape, the most important angle in construction. Obtuse (90° < θ < 180°) — wider than a right angle. Straight (θ = 180°) — a flat line. Reflex (180° < θ < 360°) — greater than a straight angle, wrapping past the half-turn. Full rotation (θ = 360°) — a complete circle back to the starting position. The calculator automatically classifies the input angle.
Complementary and Supplementary Angles
Complementary angles add up to 90°. If θ = 35°, its complement is 55°. In a right triangle, the two non-right angles are always complementary. Supplementary angles add up to 180°. If θ = 120°, its supplement is 60°. Adjacent angles on a straight line are supplementary. The reflex angle (360° − θ) completes the full rotation. These relationships are fundamental in geometry proofs, construction, and engineering. Our calculator displays all three values when applicable.
Trigonometric Functions
Every angle has associated trigonometric values: sin(θ), cos(θ), and tan(θ), along with their reciprocals csc, sec, and cot. These functions relate angles to ratios of sides in right triangles and extend to all angles via the unit circle. Key values: sin(30°)=0.5, cos(60°)=0.5, sin(45°)=cos(45°)=√2/2≈0.707, sin(90°)=1, cos(0°)=1, tan(45°)=1. Our calculator displays sin, cos, and tan for any input angle, computed using the normalized angle between 0° and 360°.
Other Angle Units: Gradians, Turns, and DMS
Gradians (also called gon or grads) divide a right angle into 100 parts, making a full circle 400 gradians. Used primarily in surveying and some European countries. 1 gradian = 0.9°. Turns (revolutions) express angles as fractions of a full rotation: 1 turn = 360° = 2π rad. Useful in rotational mechanics. DMS (degrees-minutes-seconds) subdivides degrees into 60 minutes (') and each minute into 60 seconds ("): 45.5° = 45° 30' 0". Used extensively in cartography, navigation, GPS coordinates, and astronomy. Our calculator converts to all formats simultaneously.
Angles in Engineering and Architecture
Angles are critical in structural engineering: roof pitch is specified as an angle (common pitches: 18.4°, 26.6°, 33.7°, 45°). Road gradients use angles: a 6% grade equals arctan(0.06) ≈ 3.43°. In mechanical engineering, cam profiles, gear teeth, and cutting tool geometry all depend on precise angle calculations. CNC machining requires tool angles to fractions of a degree. In architecture, angles determine sight lines, shadow projections, and aesthetic proportions. The golden angle (137.508°) appears in nature and is used in optimal design layouts.
Angles in Navigation and Astronomy
Navigation relies on angles measured in degrees: compass bearings range from 0° (north) through 90° (east), 180° (south), 270° (west). GPS coordinates use DMS or decimal degrees for latitude and longitude. Aircraft headings, ship courses, and satellite orbits are all specified as angles. In astronomy, angles measure celestial positions: right ascension (hours, analogous to longitude) and declination (degrees, analogous to latitude). Angular diameter describes how large objects appear: the Moon’s angular diameter is about 0.5°, and one degree on the sky at arm’s length is roughly the width of a finger.
Angles in Programming and Game Development
In computer science, angles are essential for graphics, robotics, and physics simulations. Most programming languages measure angles in radians: JavaScript’s Math.sin(), Python’s math.sin(), and C’s sin() all expect radian input. Game engines like Unity use degrees for inspector values but radians internally. CSS transforms use degrees (rotate(45deg)), while SVG and Canvas use radians. Quaternions, used for 3D rotation, avoid gimbal lock problems that occur with Euler angles. Understanding the relationship between degrees and radians prevents bugs in rotation calculations, physics engines, and procedural animation systems.
How to Use This Calculator
Select your input unit (degrees, radians, gradians, or turns). Enter the angle value and click calculate. The interactive protractor canvas shows your angle with labeled rays, arc, and degree markers. The angle type card classifies it automatically. All unit conversions display: degrees, radians, gradians, turns, DMS, and π-fraction. The trigonometric panel shows sin, cos, and tan. Complementary, supplementary, and reflex angles are computed. The step-by-step section details each conversion. The reference table lists standard angles from 0° to 360° with your input highlighted.