Angle Parts
Vertex: shared endpoint. Initial side: stationary ray. Terminal side: rotated ray.
What Is an Angle?
In geometry, an angle forms when two rays meet at a point, known as the vertex. The measure of rotation required to move one ray onto the other defines the angle.
Notation
Common forms include ∠ABC (B is the vertex), ∠θ, or by arrow notation showing the direction of rotation.
Units
Degrees (°), radians, and gradians. One full turn equals 360°, 2π radians, or 400 gradians.
Who Discovered Angles?
Angles evolved over millennia. Ancient Mesopotamian astronomers divided circles into 360 parts for celestial calculations. Greek mathematicians like Euclid provided the first rigorous definitions in Elements. Later, trigonometry pioneers Hipparchus and Ptolemy used angles for star charts. Islamic scholars preserved and expanded this knowledge, while Renaissance mathematicians formalized radians, linking angles to arc length and calculus.
- c. 1800 BCE: Babylonian clay tablets document 360-degree circle divisions.
- 300 BCE: Euclid describes angle postulates and theorems in Elements.
- 2nd century CE: Ptolemy’s Almagest compiles trig tables for astronomy.
- 9th–11th centuries: Islamic mathematicians refine sine and tangent tables for navigation.
- 17th century: Newton, Leibniz, and Euler popularize radian measure in calculus.
Types of Angles
Acute Angle
Measures less than 90°. Common in roof trusses, typography, and origami folds.
Right Angle
Exactly 90°. Defines perpendicular directions—essential in architecture and UI design.
Obtuse Angle
Between 90° and 180°. Appears in ergonomic furniture, satellite dishes, and animation rigs.
Straight Angle
Exactly 180°. Represents a half-turn or straight line—used in surveying and bearings.
Reflex Angle
Between 180° and 360°. Helps model full joint rotations in robotics and biomechanics.
Full Rotation
360° turn, returning to the starting orientation. Impacts navigation and rotational symmetry.
How to Measure Angles
Angles can be measured using physical tools or computed through trigonometry.
Protractor
Align the center on the vertex, place one ray along the zero line, and read the degree marking of the other ray.
Compass & Straightedge
Construct congruent angles or bisect existing ones by drawing arcs and intersection points.
Digital Tools
CAD software and smartphone apps provide high-precision angular measurements and conversions.
Trigonometric Ratios
Use sine, cosine, and tangent—plus inverse trig functions—to obtain angles from side lengths.
Angle Formulas and Theorems
- Triangle interior sum: 180° for every triangle.
- Polygon interior sum: (n − 2) × 180° for an n-sided polygon.
- Exterior angles: one per vertex, totaling 360°.
- Vertical angles: equal when two lines intersect.
- Alternate interior angles: equal when a transversal crosses parallel lines.
- Complementary / supplementary: pairs adding to 90° / 180°.
Angles in the Real World
Angles stretch far beyond math class:
- Engineering: beam forces, robotic arm movement, optical instruments.
- Architecture: framing, façade geometry, daylight analysis.
- Art & Design: perspective drawing, mandalas, logo grids.
- Sports science: shot arcs in basketball, golf swing planes, climbing routes.
- Navigation: compass bearings, aviation headings, GPS triangulation.
- Physics: wave interference, simple harmonic motion, angular momentum.
Angle FAQ
What is the standard unit for angles?
Degrees and radians are standard. Engineers often switch between them depending on context.
Who invented the degree?
Babylonian astronomers created the 360-degree division; no single inventor is known.
How do I convert degrees to radians?
Multiply the degree value by π/180. Example: 60° × π/180 = π/3 radians.
What’s the difference between acute and obtuse angles?
Acute angles are less than 90°, obtuse angles fall between 90° and 180°.
Why do complementary angles matter?
Their 90° sum is vital for constructing perpendicular lines and solving right-triangle problems.