Central Angle Calculator

What Is a Central Angle?

In plane geometry, a central angle is an angle whose apex (vertex) is located at the center $O$ of a circle and whose two sides (rays) are radii that intersect the circle's circumference at two distinct points. The central angle subtends an arc along the perimeter and encloses a region known as a circular sector.

Central angles play a fundamental role in trigonometry, geometry, navigation, mechanical gear design, and circular kinematics. Whether you know the arc length, chord length, radius, or sector area, this calculator gives you exact solutions in degrees, radians, gradians, and fractional turns.

Central Angle Formulas

Depending on which circle dimensions are known, the central angle $\theta$ can be derived using the following geometric formulas:

• From Arc Length ($s$) and Radius ($r$): $$\theta = \frac{s}{r} \text{ (radians)} = \left(\frac{s}{r}\right) \times \left(\frac{180^\circ}{\pi}\right) \text{ (degrees)}$$
• From Chord Length ($c$) and Radius ($r$): $$\theta = 2 \arcsin\left(\frac{c}{2r}\right)$$
• From Sector Area ($A$) and Radius ($r$): $$\theta = \frac{2A}{r^2} \text{ (radians)}$$
• From Arc Length ($s$) and Chord Length ($c$): Solved via the root relation $\frac{\sin(\theta/2)}{\theta/2} = \frac{c}{s}$, yielding both $\theta$ and $r = \frac{s}{\theta}$.

Associated Circle and Sector Dimensions

Once the central angle $\theta$ (in radians) and radius $r$ are known, other important circular quantities are calculated as follows:

• Arc Length ($s$): $s = r \cdot \theta$
• Chord Length ($c$): $c = 2r \sin\left(\frac{\theta}{2}\right)$
• Sector Area ($A_{\text{sec}}$): $A_{\text{sec}} = \frac{1}{2} r^2 \theta$
• Circular Segment Area ($A_{\text{seg}}$): $A_{\text{seg}} = \frac{1}{2} r^2 (\theta - \sin\theta)$
• Circle Circumference: $C = 2\pi r$
• Full Circle Area: $A = \pi r^2$

Related Circle and Geometry Calculators

Explore more geometry calculators on OnlineMiniTools:

• Arc Length Calculator - Find arc lengths, sector angles, and radii.
• Chord Length Calculator - Compute chord lengths and sagittal heights of circular segments.
• Angle Converter - Convert angles between degrees, radians, minutes, and seconds.
• Area of a Circle Calculator - Calculate circle area, radius, diameter, and perimeter.