An angle is the figure formed by two line segments extending from one point, the vertex of the angle.
If the two line segments are perpendicular, one speaks of a right angle.
In order to measure an angle, a circle centered at the vertex is drawn. The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. The degree measure of the angle is the length of the arc, divided by the circumference of the circle, and multiplied by 360.
A right angle therefore has a measure of π/2 radians and 90 degrees. Angles smaller than a right angle are called acute and angles larger than a right angle are called obtuse.
Mathematicians generally prefer angle measurements in radians because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians. The SI system of units uses radians as the (derived) unit for angles.
In the Euclidian plane, the angle θ between two vectors u and v is given by the formula
- u · v = cos(θ) ||u|| ||v||
This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>.
The angle of two intersecting curves is defined to be the angle between the tangents at the point of intersection.
See also solid angle.
The Angles are a tribe of northern Germanic descent famous for invading Britain together with the Saxons and Jutes in the 400s AD. The name England comes from "land of the Angles". This name Angle derives from an Old English word angul (lit. fishhook) that apparently described the shape of the British island. It is cognate with the word angle in mathematics described above rather than the word "angel".