Enter a radius and central angle to calculate the arc length and sector area of a circle. Choose your angle unit — degrees, radians, or gradians — and get the arc length, chord length, sector area, and full circumference all at once.
Arc Length
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Sector Area
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Chord Length
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Full Circumference
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Angle in Radians
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The arc length formula is L = r × θ, where r is the radius and θ is the central angle in radians. If your angle is in degrees, convert it first: θ (radians) = θ (degrees) × π / 180. The result gives the distance along the curved edge of the circle sector.
When the angle is already in radians, simply multiply the radius by the angle: L = r × θ. For example, a circle with radius 5 and a central angle of π/2 radians has an arc length of 5 × (π/2) ≈ 7.854 units.
If you know the arc length and the central angle, you can rearrange the formula to find the radius: r = L / θ (with θ in radians). Alternatively, if you know the diameter, the radius is simply half the diameter.
If you know the arc length and the radius, you can find the central angle using θ = L / r (in radians). You can also determine the angle from other information like the chord length or sector area using trigonometric relationships.
The standard arc length formula L = r × θ requires the angle to be in radians. However, you can work in degrees by using the equivalent formula L = (θ° / 360) × 2πr, which represents the fraction of the full circumference. This calculator handles the conversion automatically for degrees and gradians.
Arc length is the distance measured along the curved edge of the circle between two points. Chord length is the straight-line distance between those same two endpoints. The chord is always shorter than the arc, and they are equal only when the angle approaches zero.
The sector area is the 'pie slice' area enclosed by two radii and the arc. It is calculated as A = (1/2) × r² × θ (with θ in radians), or equivalently A = (1/2) × r × L, where L is the arc length. So the sector area is simply half the product of the radius and the arc length.
Gradians (also called gon) are a unit of angle measurement where a full circle equals 400 gradians. This means 100 gradians = 90 degrees = π/2 radians. Gradians are sometimes used in surveying and civil engineering. To convert gradians to radians, multiply by π/200.