Central Angle Calculator
Calculate the central angle from arc length and radius (θ = s/r) or from inscribed angle (central = 2 × inscribed).
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Frequently Asked Questions
What is a central angle?
A central angle is an angle whose vertex is at the center of the circle, formed by two radii. Its measure equals the arc it intercepts.
How do you find central angle from arc length?
Use the formula θ = s/r (in radians), where s is arc length and r is radius. To convert to degrees, multiply by 180/π.
How do you find central angle from inscribed angle?
The central angle is always twice the inscribed angle that intercepts the same arc: Central = 2 × Inscribed.
What is the difference between radians and degrees?
Radians are based on arc length (θ = s/r), while degrees divide a circle into 360 parts. One full circle = 2π radians = 360°.
Why use radians instead of degrees?
Radians provide a natural relationship between angle, radius, and arc length (s = rθ). This makes many formulas simpler and more elegant.
What is the central angle of a semicircle?
A semicircle has a central angle of 180° or π radians, which is exactly half of a full circle (360° or 2π radians).
Can a central angle be greater than 360°?
In basic geometry, central angles are typically between 0° and 360°. Angles greater than 360° represent multiple rotations around the circle.
Where are central angle calculations used?
Central angles are used in navigation, astronomy, engineering, computer graphics, clock design, and anywhere circular motion or sectors are analyzed.