Arc Length Calculator

About the Arc Length Calculator

The arc length calculator lets you quickly compute the length of an arc, the area of the corresponding sector, the straight-line chord, and the arc-to-chord ratio for any circle. An arc is a portion of the circumference of a circle defined by a radius and a central angle. The formula is beautifully simple — arc length equals the radius multiplied by the angle in radians — yet it underpins countless real-world applications from road-curve design to satellite orbits. Engineers rely on arc length when laying out highway curves, architects use it for dome and arch calculations, and machinists need it for gear-tooth profiles. This calculator goes beyond the basic formula: it also computes sector area (the "pizza slice" region), chord length (the straight line connecting the arc endpoints), and the ratio of arc to chord, which approaches 1 for small angles and diverges for larger arcs. Choose between degrees and radians, pick from common preset angles like 30°, 45°, 60°, 90°, or 180°, and see a full property breakdown with an interactive arc diagram. Whether you are a student verifying homework, a surveyor measuring curves, or a programmer implementing circular interpolation, the page keeps the related measurements together.

When This Page Helps

While the arc length formula (s = rθ) is simple, real-world problems usually require several related measurements — sector area, chord length, sagitta, and segment area — not just the arc itself. This calculator computes those values together so you can check the whole circle geometry from one radius-and-angle setup.

That is useful both for practical design work and for study. Instead of recomputing each derived quantity separately, you can see how changing the angle affects all the related measures at once.

How to Use the Inputs

1. Enter the radius of your circle in the Radius field.
2. Enter the central angle value in the Central Angle field.
3. Select Degrees or Radians as the angle unit from the dropdown.
4. Optionally choose a length unit (cm, m, in, ft) for labeled output.
5. Click a preset button like "90°, r=10" to load a common example.
6. Review arc length, sector area, chord length, sagitta, and other results below.
7. Adjust the Decimal Precision field to control the number of decimal places shown.

Example Calculation

Tips & Best Practices

• Check that all inputs use the same scale and assumptions before trusting the result.
• Compare the answer with the worked example or a rough estimate to catch entry mistakes.

Understanding Arc Length and Sector Geometry

An arc is a portion of the circumference of a circle, defined by a central angle θ and the radius r. The arc length formula s = rθ (with θ in radians) is one of the most elegant relationships in geometry — it directly ties the linear measurement of a curve to the angle that generates it. When working in degrees, convert first: θ_rad = θ_deg × π/180. The sector area formula ½r²θ gives the area of the "pizza slice" enclosed by the arc and two radii, while the chord length 2r sin(θ/2) measures the straight-line shortcut between the arc's endpoints.

Real-World Applications

Arc length calculations are essential in civil engineering for designing highway curves and railroad bends, where the radius of curvature determines safe speed limits. Architects use sector geometry when designing arched windows, domes, and amphitheaters. In manufacturing, gear tooth profiles are based on involute curves derived from arc length relationships. Satellite navigation relies on great-circle arc lengths on the Earth's surface, and computer graphics use arc interpolation for smooth animation paths.

Segment, Sagitta, and Chord Relationships

The segment area (the region between a chord and its arc) equals the sector area minus the triangle area: A_segment = ½r²(θ − sin θ). The sagitta (or versine) h = r − r cos(θ/2) is the maximum height of the arc above the chord. These measurements are critical in bridge and tunnel design, where the sagitta determines the rise of an arch. The arc-to-chord ratio approaches 1 for small angles and increases for larger arcs, providing a useful check on whether a curved path is significantly longer than the straight-line alternative.

Frequently Asked Questions

• What is arc length?

  Arc length is the distance measured along a curved line forming part of the circumference of a circle. It depends on the radius and the central angle.

• How do I convert degrees to radians?

  Multiply the angle in degrees by π/180. For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 radians.

• What is the difference between arc length and chord length?

  Arc length is the curved distance along the circle, while chord length is the straight-line distance between the two endpoints of the arc.

• When does the arc-to-chord ratio equal 1?

  The ratio approaches 1 as the central angle approaches 0, because the curve becomes indistinguishable from the straight line for very small angles.

• What is a sector area?

  Sector area is the region enclosed by two radii and the arc, like a "pizza slice". It equals ½ r² θ with θ in radians.

• Can I use this for angles greater than 360°?

  Yes. An angle greater than 360° simply means the arc wraps around the circle more than once; the formulas still apply.