Central Angle Calculator
Calculate the central angle of a circle in degrees and radians. Choose your known values — arc length, chord length, sector area, or arc percentage — and the tool returns the angle and related circle measures.
any unit
same unit as r
θ (rad) = L / r
Central angle formulas used
From arc length L and radius r: \(\theta = \frac{L}{r}\) (radians)
From chord length c and radius r: \(\theta = 2 \arcsin\left(\frac{c}{2r}\right)\)
From sector area A and radius r: \(\theta = \frac{2A}{r^2}\)
Degrees ↔ radians: \(\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}\), \(\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}\)
How to use this central angle calculator
- Select the tab that matches what you know (arc, chord, sector, percent).
- Enter your values using the same unit for all lengths.
- Click Calculate to get the angle in both degrees and radians.
- Use the extra values (arc, chord, sector area) in your geometry or construction problem.
This page is meant to centralize the 4 most common central-angle problems in one place, so you don’t have to switch tools like on many competitors’ sites.
FAQs
What happens if the chord is longer than the diameter?
That’s not possible in a circle — the calculator will warn you because c ≤ 2r must hold.
Can I get 360°?
Yes. If the arc length equals the full circumference (2πr) or the percent is 100, you get 360°.
Why do we often use radians here?
Because arc length and sector area formulas are the cleanest in radians: L = rθ and A = ½ r² θ.
Formula (LaTeX) + variables + units
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From arc length L and radius r: \(\theta = \frac{L}{r}\) (radians) From chord length c and radius r: \(\theta = 2 \arcsin\left(\frac{c}{2r}\right)\) From sector area A and radius r: \(\theta = \frac{2A}{r^2}\) Degrees ↔ radians: \(\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}\), \(\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}\)
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.