Inscribed Angle Calculator
Calculate inscribed angles, central angles, and arc lengths using the inscribed angle theorem
Calculate Inscribed Angles
Units
Angle at the center of the circle
Required for arc length calculations
Inscribed Angle Results
Angle Measurements
Inscribed Angle Theorem
Theorem: The inscribed angle is half the central angle
Formula: θi = θc/2
Arc Length: L = θc × r (θc in radians)
From Inscribed Angle: L = 2 × θi × r
Central from Arc: θc = L/r
Example Calculation
Example: Inscribed Angle in a Circle
Problem: Find the inscribed angle when the central angle is 120°
Given: Central angle θc = 120°
Solution
Step 1: Apply the inscribed angle theorem: θi = θc/2
Step 2: θi = 120°/2 = 60°
Step 3: The inscribed angle is 60°, which is half the central angle
Note: This relationship holds regardless of where the vertex is on the circumference
Inscribed Angle Properties
Inscribed Angle
Formed by two chords meeting on the circle
Always half the central angle
Central Angle
Formed by two radii from the center
Twice the inscribed angle
Arc Length
Length of the curve between endpoints
L = central angle × radius
Diameter Property
Angle inscribed in semicircle
Always equals 90°
Circle Geometry Tips
Inscribed angle is always half the central angle
All inscribed angles subtending the same arc are equal
Angle inscribed in semicircle is always 90°
Moving vertex on circle doesn't change inscribed angle
Understanding the Inscribed Angle Theorem
What is an Inscribed Angle?
An inscribed angle is formed by two chords that intersect on the circumference of a circle. The vertex of the angle lies on the circle, and the sides of the angle are chords that extend to two other points on the circle.
Real-World Applications
- •Architecture and dome construction
- •Navigation and GPS calculations
- •Satellite positioning and astronomy
- •Computer graphics and game development
- •Engineering design and wheel mechanics
The Inscribed Angle Theorem
Main Theorem: θi = θc/2
Arc Length: L = θc × r (radians)
From Inscribed: L = 2 × θi × r
Special Case: Diameter inscribes 90°
Key Principles
- • Central angle is twice the inscribed angle
- • Same arc produces equal inscribed angles
- • Vertex position on circle doesn't matter
- • Diameter always creates 90° inscribed angle
- • Useful for solving complex circle problems
Note: This theorem is fundamental in circle geometry and appears in many geometric proofs