What is a central angle?

A central angle is an angle whose vertex is at the center of a circle and whose sides intercept an arc. It directly determines the arc length and sector area.

How do I find the central angle from arc length?

Use θ = L / r, where L is arc length and r is radius. θ is in radians. To get degrees, multiply by 180/π.

How do I find the central angle from chord length?

Use θ = 2 · arcsin(c / (2r)), where c is the chord length and r is the radius.

Can this calculator also give arc length and sector area?

Yes. Once the central angle is known, we compute arc length, sector area, and the corresponding chord length.

Central Angle Calculator

Central angle calculator for circles. Find the angle from arc length and radius, from chord length and radius, from sector area, or from arc percentage. Shows the angle in degrees and radians and returns arc, chord, and sector area values.

Select known values

Choose the scenario that matches your problem and enter the matching inputs. The calculator handles arc length, chord length, sector area, or arc percent values.

Radius (r)

Any unit.

Arc length (L)

Same unit as radius.

θ (rad) = L / r

Central angle (largest value)

—

Degree equivalent shown below.

Degrees —

Radians —

Fraction of circle —

Arc length: —

Chord length: —

Sector area: —

How to Use This Central Angle Calculator

Pick the tab that matches the values you already know, enter those values using the same units, and then click Calculate. The results card updates with the angle (both degrees and radians), how big that angle is compared to the full circle, and the related arc, chord, and sector area if a radius is provided.

Methodology

The engine uses the classic circle relationships: L = rθ, c = 2r sin(θ/2), and A = ½ r² θ. For the percent tab we reverse the fraction of the circle to degrees, then convert to radians. All angles are rounded consistently to four decimal places so the UI produces deterministic output across browsers.

Results are estimates meant for planning and educational use. They mirror the same formulas that lenders and engineers apply, but always double-check your values with more specialized tools when precision matters.

Full original guide (expanded)

The original content described the four central-angle workflows, followed by detailed formulas, how-to steps, and a short FAQ. Those explanations now live here so nothing is discarded.

Tabs at a glance

Arc length + radius: Enter the radius and arc length to compute the angle via θ = L / r.
Chord + radius: Use the radius and chord to calculate the angle with θ = 2 · arcsin(c / (2r)).
Sector area + radius: Apply θ = 2A / r² after supplying the radius and sector area.
Arc percent: Specify the percentage of the circle and the calculator converts it into degrees (θ = % × 360 / 100) and radians.

FAQs

What happens if the chord is longer than the diameter?

In that case the chord cannot exist inside the circle because c must satisfy c ≤ 2r. The calculator warns you when the input exceeds the diameter.

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?

Arc length and sector area formulas are cleanest in radians (L = rθ and A = ½ r² θ), so we keep the core math in radians even while showing degrees for readers.

Related circle tools

Tip

For engineering drawings, this calculator gives you the precise central angle to plug into CAD when you already know the arc length and the radius.

Formulas

Arc length: θ = L / r (radians)

Chord length: θ = 2 · arcsin(c / (2r))

Sector area: θ = 2A / r²

Degrees ↔ radians: θ° = θ × 180/π, θ = θ° × π/180

Citations

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/

Changelog

0.1.0-draft — 2026-01-19: Initial draft (review required).

✓ Verified by Ugo Candido Last Updated: 2026-01-19 Version 0.1.0-draft

Version 1.5.0