Calculate the central angle of a circle from arc length, chord length, or sector area. Essential for geometry, trigonometry, and circular design.
Last updated: April 2026 | By Patchworkr Team
A central angle is an angle whose vertex is at the center of a circle, with its two sides (radii) extending to the circle's circumference. The central angle intercepts an arc and determines both the arc length and the sector area.
Key relationships:
Determine the radius of the circle. This is the distance from the center to any point on the circumference. Why: The radius is the fundamental parameter. All formulas for central angles depend on it, and changing the radius changes the chord length and arc length for the same angle.
Decide what you know: arc length, chord length, or sector area. Each method uses a different approach. Why: In real applications, you might measure a chord at a construction site, or know the arc length from a blueprint, or have a sector area from a design specification. Different situations require different inverse formulas.
Why: These formulas are mathematical inverses of the primary relationships. They reverse-engineer the angle from measurements. All three formulas are exact and equivalent—they just approach the problem from different starting points.
Perform the calculation to get θ in radians. Radians are the natural unit for angle in calculus and physics because they relate directly to arc length via θ = s/r. Why: Radians are dimensionless (they're a ratio) and simplify mathematical analysis. Many applications prefer radians, and conversion to degrees is a final step.
Convert the angle to degrees if needed using θ(degrees) = θ(radians) × 180/π. Verify the result is reasonable: 0° to 360° for a standard circle, and check that all other properties (arc, chord, area) are consistent. Why: Degrees are more intuitive for most people, and cross-checking using multiple formulas ensures your answer is correct before using it in design or construction.
Pizza Slice Analysis
An angle formed at the center of a circle by two radii. It intercepts an arc and determines the arc length and sector area.
Divide arc length by radius: θ = s/r (result in radians). Convert to degrees: multiply by 180/π.
A central angle's vertex is at the center. An inscribed angle's vertex is on the circle. An inscribed angle is half the central angle intercepting the same arc.
In standard geometry, no—a full circle is 360°. But in calculus and physics, angles can exceed 360° to represent multiple rotations.
Chord length c = 2r·sin(θ/2). As the angle increases, the chord lengthens, reaching maximum (diameter = 2r) at 180°.
Radians are an angle unit where a full circle = 2π. They simplify formulas: arc length = rθ (no π needed). 1 radian ≈ 57.3°.
Once you have the central angle θ (in radians), use A = ½r²θ. Or as a fraction of the circle: A = (θ/2π) × πr².
The arc length formula s = rθ requires θ in radians. For degrees, use s = (θ × π × r) / 180.
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