Angle
GeometryAn angle is the figure formed by two rays that share a common endpoint, measured in degrees or radians.
Formula
\text{Full rotation} = 360^\circ = 2\pi \text{ radians}
Definition
An angle is formed when two rays start at the same point. The shared starting point is called the vertex and the two rays are the sides of the angle. We measure angles in degrees to describe how wide open they are.
Example
When you open a book, the two covers form an angle. The corner of a square is a $90^\circ$ angle. A full circle is $360^\circ$. The hands of a clock at 3 o'clock form a $90^\circ$ angle.
Key Insight
Angles measure rotation or "openness." Degrees divide a full turn into $360$ equal parts - a choice made by ancient Babylonian astronomers. That is why there are $360^\circ$ in a circle.
Definition
An angle is formed by two rays (the sides) sharing a common endpoint (the vertex). The measure of an angle describes the amount of rotation from one side to the other. Angles are measured in degrees ($1/360$ of a full rotation) or radians (where a full rotation $= 2\pi$ radians).
Example
Angle $ABC$ has vertex at $B$, with rays $BA$ and $BC$ as its sides. If arc measure $= 75^\circ$, the angle is acute. To convert: $90^\circ = \pi/2$ radians; $180^\circ = \pi$ radians.
Key Insight
Radians are the natural unit: the radian measure of an angle equals the arc length on a unit circle subtended by that angle. This makes radian measure dimensionless and essential in calculus (derivatives of trig functions use radians).
Definition
An angle in the plane is a rotation between two rays from a common vertex, measured as the ratio of arc length to radius on any circle centered at the vertex (radian measure). In directed angle terms, angles are equivalence classes of rotations modulo $2\pi$. In linear algebra, the angle between vectors $u$ and $v$ satisfies $\cos\theta = (u \cdot v) / (|u| |v|)$.
Example
The angle between vectors $u = (1,0)$ and $v = (1,1)$ is $\arccos((1\cdot1 + 0\cdot1) / (1 \cdot \sqrt{2})) = \arccos(1/\sqrt{2}) = 45^\circ = \pi/4$ radians.
Key Insight
The dot product formula for angles extends to any inner product space, defining "angle" in abstract settings like function spaces. This generalization underpins Fourier analysis, where orthogonality of functions is analogous to perpendicularity of vectors.