Circle Sector Calculator
Sector area, arc length and chord of a circle.
Sector area
52.3599
60° of radius 10
Arc length
10.472
Chord length
10
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How the Circle Sector Calculator works
The Circle Sector Calculator computes the area, arc length, and chord length of any circular sector — the 'pie slice' shape defined by a central angle and a radius — making it an essential tool for students, engineers, architects, and anyone working with circular geometry.
A circle sector is the region enclosed by two radii and the arc between them, essentially a pie slice cut from a full circle. The two defining inputs are the radius (r) and the central angle (θ), which can be expressed in either degrees or radians. The calculator uses these two values to derive all three key measurements: the sector area, the arc length along the curved boundary, and the straight-line chord connecting the two endpoints of the arc. Understanding which angle unit you're working in is the single most common source of error — always confirm whether your angle is in degrees or radians before calculating.
Sector area represents the two-dimensional space enclosed within the pie slice. It scales with the square of the radius, meaning doubling the radius quadruples the area for the same angle. Arc length, by contrast, is a one-dimensional measurement along the curved edge of the sector and scales linearly with the radius. The chord is the straight line connecting the two points where the radii meet the circle's circumference; it is always shorter than the arc length (except when the angle is zero) and represents the 'base' of the triangle-like shape formed inside the sector. Recognizing the geometric distinction between these three outputs prevents confusion when applying results in real-world contexts like pipe bending, land surveying, or pizza portioning.
The central angle is the most influential single variable in all three formulas. A sector with a 360° angle is simply the full circle, so its area equals πr² and its arc length equals the full circumference 2πr. A 180° sector is a semicircle, and so on proportionally. Because of this linear relationship with angle (for arc length and area), you can use the calculator as a quick cross-check: if you double the central angle while keeping the radius fixed, the arc length and area should also double. The chord, however, does not scale linearly with angle — it follows a sine relationship — so it requires the full formula and cannot be estimated by simple proportion.
A common practical mistake is confusing the arc length with the chord length when measuring physical objects. If you are bending a metal strip around a curve, you need the arc length (the actual material length). If you are cutting a straight brace across the opening of the sector, you need the chord length. Another frequent error occurs when working with very large or very small angles: near 0° the chord and arc are nearly equal, while near 360° the chord approaches zero again even though the arc is nearly the full circumference. Always sanity-check results against these boundary behaviors to catch unit or input errors early.
Formula
Area = ½r²θ · Arc = rθ (θ in radians)
Pro tips
- Always confirm your angle unit before entering values — using degrees in a radians-based formula will produce results that are off by a factor of π/180, a subtle error that is easy to miss.
- Use the arc length output (not the chord) when calculating the actual material length needed for curved physical objects like bent pipes, garden edging, or decorative arches.
- To find a missing radius or angle, work backward: if you know the arc length and angle, radius = arc length ÷ θ (in radians); if you know the area and radius, θ = 2 × area ÷ r².
- When the sector angle exceeds 180°, the chord length starts to decrease even as the arc continues to grow — keep this counterintuitive behavior in mind when designing structures with large sweep angles.
- For quick estimation, remember that a 90° (quarter-circle) sector has an area exactly one-quarter of πr² and an arc length one-quarter of 2πr — use these reference points to sanity-check any calculator output.
Key terms
- Sector
- — A sector is the pie-slice region of a circle bounded by two radii and the arc between them, defined completely by a radius and a central angle.
- Central Angle
- — The central angle is the angle formed at the center of the circle between the two radii that define the sector, measured in degrees or radians.
- Arc Length
- — Arc length is the distance along the curved edge (the circumference portion) of the sector, calculated as the product of the radius and the central angle in radians.
- Chord
- — The chord of a sector is the straight line segment connecting the two endpoints of the arc, representing the shortest path between those two boundary points.
- Radius
- — The radius is the fixed distance from the center of the circle to any point on its circumference, and it is one of the two required inputs for all sector calculations.
- Radian
- — A radian is the SI unit of angular measure where one full revolution equals 2π radians (approximately 6.2832), used directly in all core sector formulas without conversion.