An angle is a geometric figure created by two straight lines or rays that meet at a common point, called the vertex. Each of the meeting lines is a side of the angle and the region between them is the angle’s interior. Angles can also be formed by the intersection of two plane surfaces, or by rotating one ray about its vertex until it coincides with the other. Visual symbols often use lower‑case Greek letters—α, β, γ, θ—to name angles.

Notation and basic properties

Angles are named in several ways: by a single letter (commonly a Greek letter), by the vertex with a small arc mark, or by three letters indicating the two sides and the vertex in the middle (for example, ∠ABC). The amount of rotation from one side to the other measures the size of the angle. In Euclidean geometry an angle is determined by its sides and vertex; angles at corners of polygons and polyhedra are fundamental to their shape. For instance, the three edges of a triangle meet pairwise at three vertices; the edges and faces of a cube meet to form right angles at its corners.

Units and conversion

The most common units for measuring angles are degrees and radians. A full circle equals 360 degrees (360°) or 2π radians. Degrees are subdivided: 1° = 60 minutes (') and 1' = 60 seconds ("). Decimal degrees and fractional degrees are both used in practice. The radian is a pure ratio: one radian is the angle subtended by an arc whose length equals the radius of its circle. A less common unit is the gradian (also called gon), where 100 grad = 90°. Examples: 22.5° = 22°30' = π/8 rad.

Classification of angles

  • Acute angle: greater than 0° and less than 90°.
  • Right angle: exactly 90°; perpendicular lines meet at a right angle.
  • Obtuse angle: greater than 90° and less than 180°.
  • Straight angle: exactly 180°; the sides form a line.
  • Reflex angle: greater than 180° and less than 360°.
  • Full angle: exactly 360°, a complete rotation.

Angles also relate pairwise: complementary angles sum to 90°, supplementary angles sum to 180°, adjacent angles share a side, and vertical (opposite) angles are equal when two lines cross.

History and development

The division of the circle into 360 parts has ancient roots, commonly attributed to astronomical observations and sexagesimal counting systems used in Mesopotamia; this influenced the use of degrees and minutes. The radian emerged later as a natural unit in calculus and analysis because it links linear and angular measures through the circle's radius. The gradian was introduced during metrication efforts to provide a decimal subdivision of right angles. Mathematical notation and the systematic study of angles grew with Greek geometry and developed further through Islamic and European mathematics.

Applications and notable facts

Angles are central across fields: in surveying and navigation they locate directions and bearings; in engineering and carpentry they define joints and tolerances; in physics and astronomy they describe orbits and apparent separations; in computer graphics and robotics they control rotations and orientations. Trigonometry relates angles to ratios of side lengths in right triangles and underpins wave, signal and rotational analysis. In three dimensions, dihedral angles measure the angle between two planes, and solid angles generalize the concept of planar angle to a portion of space.

Practical tips: when solving geometric problems, check whether degrees or radians are required, label the vertex clearly, and use standard relationships (complementary, supplementary, vertical) to reduce construction and calculation. For further structured introductions, consult elementary geometry resources or textbooks on trigonometry and analytic geometry. For definitions and illustrations see lines, sides, point, plane, α, β, γ, θ, measurement, degrees, symbol, decimal, fraction, minutes, seconds, mathematics, radians, gradians, vertex, triangle, cube.