Angle
An angle is the measure of rotation between two rays (sides) that share a common endpoint called the vertex. Angles are measured in degrees (°) or radians (rad) and describe the amount of turn between two directions. They are fundamental to geometry, trigonometry, physics, and engineering, appearing in everything from architectural blueprints to robotic arm movements.
Key Formula
θ = arc length / radius (in radians)
LaTeX: \theta = \frac{\text{arc length}}{\text{radius}} \text{ (in radians)}
| Symbol | Meaning | Unit |
|---|---|---|
| θ | angle in radians | rad |
| s | arc length | m |
| r | radius | m |
Worked Example
Problem
Convert 135° to radians and identify the type of angle.
Solution
Step 1: Use conversion factor: radians = degrees × (π / 180). Step 2: 135 × (π / 180) = 135π / 180 = 3π / 4. Step 3: Since 90° < 135° < 180°, it is an obtuse angle.
Answer
3π/4 radians; obtuse angle
Classification of Angles by Measure
| Type | Measure | Example | Real-world Example |
|---|---|---|---|
| Zero angle | 0° | Two rays in same direction | Fully open flat surface |
| Acute angle | 0° < θ < 90° | 45°, 60° | Roof slope, ramp |
| Right angle | θ = 90° | Corner of a square | Floor meeting a wall |
| Obtuse angle | 90° < θ < 180° | 120°, 150° | Obtuse triangle corner |
| Straight angle | θ = 180° | Straight line | Flat road |
| Reflex angle | 180° < θ < 360° | 270° | Clock hand at 9:00 going backwards |
Interactive Tools
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Related Terms
Triangle
A triangle is a polygon with exactly three sides, three angles, and three vertices. The sum of the interior angles of any triangle always equals 180°, making it the simplest closed polygon. Triangles are the most rigid of all polygons and are widely used in engineering structures, architecture, and navigation due to their inherent stability.
Right Triangle
A right triangle is a triangle containing exactly one right angle (90°). The side opposite the right angle is called the hypotenuse and is always the longest side, while the other two sides are called legs or catheti. Right triangles are the foundation of trigonometry and appear throughout architecture, engineering, and physics in the analysis of forces, distances, and angles.
Line (geometry)
A line in geometry is a one-dimensional figure that extends infinitely in both directions and has no endpoints, width, or curvature. It is defined by any two distinct points on it and is the shortest path between those points when considered in a straight path. Lines are foundational to Euclidean geometry and are used to construct angles, polygons, and coordinate systems.
From Latin "angulus" meaning a corner or angle, related to "angere" (to bend). Greek used "gonia" (γωνία) for angle, which appears in words like "polygon" and "trigonometry". The word "angle" has been in English since the 14th century through Old French.