What is the formula for the central angle of a circle?

The central angle in radians equals the arc length divided by the radius: theta = arc length / radius. To convert to degrees, multiply by 180 and divide by pi: degrees = (arc length / radius) x (180 / pi). Alternatively, if you know the sector area and radius, the central angle in radians is: theta = (2 x sector area) / radius squared.

How do I find the central angle from the arc length and radius?

Divide the arc length by the radius to get the central angle in radians. For example, if the arc length is 7.854 and the radius is 5, the central angle is 7.854 / 5 = 1.5708 radians, which equals 90 degrees. The relationship comes from the definition of a radian: one radian is the angle subtended when the arc length equals the radius.

What is the difference between a central angle and an inscribed angle?

A central angle has its vertex at the centre of the circle, and its two sides (radii) extend to the circumference. An inscribed angle has its vertex on the circumference, with its two sides as chords of the circle. The inscribed angle theorem states that an inscribed angle is exactly half the central angle that subtends the same arc. So if a central angle is 90 degrees, an inscribed angle subtending the same arc is 45 degrees.

Central Angle Calculator

Calculate the central angle of a circle from the arc length and radius, or from the sector area and radius. Results are shown in both degrees and radians, together with the full set of circle sector measurements.

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Standard formula Central angle = arc length / radius (radians). Sector area = ½r²θ.

Input Method

Arc length (any unit)

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Radius (same unit)

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Sector area (square units)

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Radius (same unit)

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Angle Unit Display

Primary angle unit for results

Decimal places

Central Angle Results

Central Angle
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degrees

Central Angle

radians

Arc Length

units

Sector Area

square units

Full Sector Measurements

Radius (r)-

Central angle (θ)-

Arc length (s)-

Sector area (A)-

Chord length-

Sector perimeter-

Formula Verification

Method used-

Formula-

Working-

Full circle circumference-

Arc as fraction of circle-

Check: arc = r x θ-

Worked Example (default inputs)

Given arc length = 7.854 units and radius = 5 units:
Central angle (θ) = arc length / radius = 7.854 / 5 = 1.5708 radians
Convert to degrees: 1.5708 × (180 / π) = 90.000°
Sector area = ½ × r² × θ = 0.5 × 25 × 1.5708 = 19.635 square units
Chord length = 2r × sin(θ/2) = 10 × sin(45°) = 7.071 units
Sector perimeter = arc + 2r = 7.854 + 10 = 17.854 units

What Is a Central Angle?

A central angle is an angle formed at the centre of a circle, with its two sides being radii that extend to the circumference. The arc of the circle that lies between those two radii is called the intercepted arc. The central angle and its intercepted arc have the same angular measure: a central angle of 90 degrees subtends exactly one quarter of the circle's circumference.

Central angles are fundamental to circle geometry and appear in many practical applications: from calculating the area of a pie slice (sector), to describing the sweep of a sprinkler, to finding the angle subtended by a road curve.

Central Angle Formulas

There are two standard ways to calculate the central angle, depending on what information you have.

Known values	Formula (radians)	Formula (degrees)
Arc length (s) and radius (r)	θ = s / r	θ = (s / r) × (180 / π)
Sector area (A) and radius (r)	θ = 2A / r²	θ = (2A / r²) × (180 / π)
Fraction of circle (f)	θ = 2πf	θ = 360 × f

The arc length formula (θ = s / r) comes directly from the definition of a radian: one radian is the angle at the centre of a circle where the arc length equals the radius. This makes radian measure the natural unit for circular geometry.

Related Circle Measurements

Once you know the central angle and radius, you can find every other measurement of the circular sector:

Arc length: s = r × θ (where θ is in radians)
Sector area: A = ½ × r² × θ (where θ is in radians)
Chord length: c = 2r × sin(θ / 2)
Sector perimeter: P = s + 2r (arc plus the two radii)

Degrees vs Radians

Degrees divide a full circle into 360 equal parts. Radians relate the angle to the arc length and radius directly: a full circle is 2π radians (approximately 6.2832 radians). The conversion is simple: multiply radians by 180/π to get degrees, or multiply degrees by π/180 to get radians.

Degrees	Radians	Description
30°	π/6 ≈ 0.5236	One twelfth of a circle
45°	π/4 ≈ 0.7854	One eighth of a circle
60°	π/3 ≈ 1.0472	One sixth of a circle
90°	π/2 ≈ 1.5708	Quarter circle (right angle)
180°	π ≈ 3.1416	Half circle (straight angle)
270°	3π/2 ≈ 4.7124	Three-quarter circle
360°	2π ≈ 6.2832	Full circle

Related Calculators

Maths and Stats Calculators: the full hub for geometry and maths tools.
Arc Length and Sector Area Calculator: find the arc and sector from angle and radius.
Circumference Calculator: calculate the perimeter of a full circle.
Area of a Circle Calculator: find the area of a full circle from its radius or diameter.
Annulus Calculator: area and perimeter of a ring between two concentric circles.

Method: Standard circle geometry. Central angle from arc length: θ (rad) = s / r; convert to degrees by multiplying by 180/π. Central angle from sector area: θ (rad) = 2A / r². Chord length: c = 2r sin(θ/2). Sector perimeter: P = s + 2r. All formulas are standard Euclidean geometry.

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