Arc Length Calculator
Arc Length
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Sector Area
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Chord Length
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How Arc Length Works
Arc length is the distance along the curved portion of a circle between two points on its circumference. It is one of the fundamental measurements in geometry and appears in applications ranging from engineering to astronomy. According to standard mathematics references such as Wolfram MathWorld, the arc length formula s = r x theta (where theta is in radians) dates back to the earliest studies of circular geometry by Greek mathematicians. The formula's simplicity when using radians is, in fact, the primary reason radians are the preferred angle unit in mathematics and physics.
This calculator computes three related measurements from a radius and central angle: the arc length, the sector area (the "pie slice" region bounded by two radii and the arc), and the chord length (the straight line connecting the arc's endpoints). It accepts angles in either degrees or radians. For full-circle calculations, use our Circumference Calculator, and for the area of enclosed sectors, try the Sector Area Calculator.
The Arc Length Formula
The core formulas used by this calculator are derived from the definition of radian measure as established in standard OpenStax Precalculus textbooks:
- Arc length: s = r x theta (radians), or s = r x theta x pi/180 (degrees)
- Sector area: A = 0.5 x r² x theta (radians)
- Chord length: c = 2r x sin(theta/2)
Where r = radius, theta = central angle, and pi = 3.14159...
Worked example: A circle with radius 10 cm and a central angle of 60 degrees. Convert to radians: 60 x pi/180 = 1.0472 rad. Arc length = 10 x 1.0472 = 10.472 cm. Sector area = 0.5 x 100 x 1.0472 = 52.36 cm². Chord length = 2(10) x sin(1.0472/2) = 20 x sin(0.5236) = 20 x 0.5 = 10 cm. Notice that the chord equals the radius when the angle is exactly 60 degrees, forming an equilateral triangle.
Key Terms You Should Know
- Arc -- a continuous portion of a circle's circumference. A minor arc spans less than 180 degrees; a major arc spans more than 180 degrees.
- Central angle -- the angle formed at the center of a circle by two radii. It subtends (defines) an arc on the circumference.
- Radian -- the angle subtended when the arc length equals the radius. One full revolution = 2 pi radians = 360 degrees. One radian is approximately 57.296 degrees.
- Chord -- a straight line segment connecting two points on a circle. It is always shorter than the arc between those same two points (except when the angle is 0).
- Sector -- the "pie slice" region enclosed by two radii and the arc between them. Its area is proportional to the central angle.
Arc Length vs Chord Length Comparison
The difference between arc length and chord length grows as the central angle increases. At small angles, the arc and chord are nearly equal; at 180 degrees, the chord equals the diameter (2r) while the arc equals half the circumference (pi x r, approximately 3.14r).
| Central Angle | Arc Length (r=10) | Chord Length (r=10) | Sector Area (r=10) |
|---|---|---|---|
| 30 degrees | 5.236 | 5.176 | 26.18 |
| 60 degrees | 10.472 | 10.000 | 52.36 |
| 90 degrees | 15.708 | 14.142 | 78.54 |
| 120 degrees | 20.944 | 17.321 | 104.72 |
| 180 degrees | 31.416 | 20.000 | 157.08 |
| 270 degrees | 47.124 | 14.142 | 235.62 |
| 360 degrees | 62.832 | 0.000 | 314.16 |
Practical Examples
Example 1 -- Curved road design: A highway curve has a radius of 300 meters and sweeps through 45 degrees. Arc length = 300 x (45 x pi/180) = 300 x 0.7854 = 235.6 meters. The road surface area in the curve = 235.6 m x lane width. Civil engineers use this calculation to determine paving material quantities and guardrail lengths.
Example 2 -- Pulley belt length: A pulley with radius 8 cm has a belt wrapped around 200 degrees of its circumference. Arc length = 8 x (200 x pi/180) = 8 x 3.4907 = 27.93 cm of belt in contact with the pulley. The remaining belt runs straight to the next pulley. Use our Circle Calculator for full circumference measurements.
Example 3 -- Pizza slice: A 14-inch diameter pizza (radius = 7 inches) cut into 8 equal slices. Each slice has a central angle of 45 degrees. Arc length (crust edge) = 7 x (45 x pi/180) = 5.50 inches. Sector area = 0.5 x 49 x 0.7854 = 19.24 square inches per slice. This is the same math behind calculating material for fans, clock faces, and pie charts.
Tips for Arc Length Calculations
- Always check your angle unit. The most common mistake is using degrees in a formula that expects radians. If your angle is in degrees, either convert first (multiply by pi/180) or use the degree-specific formula s = r x theta x pi/180.
- Use radians for calculus. In calculus, arc length formulas and derivatives involving trigonometric functions only work correctly in radians. The formula s = r x theta requires theta in radians.
- For non-circular curves, use integration. The arc length formula s = r x theta only works for circular arcs. For ellipses, parabolas, or arbitrary curves, arc length requires the integral formula: s = integral of sqrt(1 + (dy/dx)²) dx. Try our Ellipse Calculator for elliptical arcs.
- Arc length is always greater than chord length. This is because the shortest distance between two points is a straight line (the chord). The only exception is when the angle is 0 degrees, where both are zero.
- A full circle is a special case. When theta = 360 degrees (2 pi radians), the arc length equals the circumference (2 pi r), the chord length is 0 (start and end points coincide), and the sector area equals the full circle area (pi r²).
Real-World Applications of Arc Length
Arc length calculations are used extensively across engineering, architecture, and science. In mechanical engineering, they determine belt and chain lengths for pulleys and gears. In civil engineering, they calculate road curve lengths, railway track bends, and bridge arch dimensions. Architects use arc length for curved walls, domed ceilings, and stadium seating layouts. In astronomy, the angular diameter of celestial objects (measured in arcseconds) combined with distance gives the actual physical size using arc length geometry. GPS navigation systems use arc length on Earth's surface (great circle distance) to calculate distances between coordinates, treating Earth as a sphere with radius 6,371 km. Even graphic design software uses arc length computations when drawing Bezier curves and circular text paths.
Frequently Asked Questions
What is the formula for arc length?
The arc length formula is s = r x theta, where r is the radius and theta is the central angle in radians. If the angle is given in degrees, convert first by multiplying by pi/180, giving s = r x theta x pi/180. For a full circle (theta = 360 degrees = 2 pi radians), the arc length equals the circumference: s = 2 x pi x r. For example, a circle with radius 5 and central angle of 90 degrees has arc length = 5 x 90 x pi/180 = 5 x 1.5708 = 7.854 units. This formula only applies to circular arcs; non-circular curves require integration.
What is the difference between arc length and chord length?
Arc length is the distance measured along the curved path of the circle, while chord length is the straight-line distance between the two endpoints. The arc is always longer than the chord because a straight line is the shortest path between two points. At small angles, the difference is negligible -- for a 10-degree arc, the chord is about 99.97% of the arc length. At 90 degrees, the chord is about 90% of the arc length. At 180 degrees, the chord (the diameter) is only about 63.7% of the arc length. The chord formula is c = 2r x sin(theta/2).
When should I use radians instead of degrees?
Radians are the natural unit for angle measurement in mathematics and physics. Use radians whenever working with calculus (derivatives and integrals of trigonometric functions only work correctly in radians), the arc length formula (s = r x theta is simplest in radians), angular velocity (omega = theta/t in radians per second), and any physics formula involving circular motion. Use degrees for everyday communication, navigation, and contexts where people are more familiar with the 360-degree system. One radian equals approximately 57.296 degrees, and one full revolution is 2 pi radians (about 6.2832).
How do I find the radius from arc length and angle?
Rearrange the arc length formula to solve for radius: r = s / theta, where s is the arc length and theta is the central angle in radians. If the angle is in degrees, use r = s / (theta x pi/180). For example, if an arc is 15 cm long and the central angle is 1.2 radians, then r = 15 / 1.2 = 12.5 cm. This inverse calculation is commonly used in surveying and construction when you measure a curved section and need to determine the radius of curvature for design specifications.
How is arc length used in real-world engineering?
Arc length is essential in multiple engineering disciplines. In road design, highway curves are specified by radius and arc length -- a typical highway exit ramp might have a 200-meter radius over 90 degrees, giving an arc length of 314 meters. In mechanical engineering, belt and chain drives use arc length to determine how much belt contacts each pulley. Gear teeth spacing relies on arc length calculations along the pitch circle. In architecture, curved walls, arched bridges, and dome segments all require arc length measurements for material estimation. CNC machining uses arc length to program tool paths along circular interpolation movements.
Can I calculate arc length for an ellipse?
Unlike circles, there is no simple closed-form formula for the arc length of an ellipse. Elliptical arc length requires computing an elliptic integral, which has no exact algebraic solution and must be evaluated numerically. For approximate calculations, Ramanujan's formula gives the full circumference of an ellipse: C is approximately pi x [3(a+b) - sqrt((3a+b)(a+3b))], where a and b are the semi-major and semi-minor axes. For partial elliptical arcs, numerical integration is necessary. Use our Ellipse Calculator for ellipse-specific calculations.