Arc Length
The arc length equals the radius multiplied by the central angle in radians.
s = rθ
Radius from Arc Length
Determines the radius of the circle given a known arc length and central angle.
r = s / θ
Central Angle from Arc Length
Finds the central angle subtended by a given arc length and radius.
θ = s / r
How It Works
An arc is a portion of a circle's circumference. The arc length formula s = rθ is one of the simplest in geometry: multiply the radius by the central angle in radians. This calculator handles degree-to-radian conversion automatically, so you can enter the angle in either unit.
Example Problem
A circle has radius 5 m and a central angle of 60°. Find the arc length.
- Identify the known values: radius r = 5 m, central angle θ = 60°.
- Determine what to solve for: the arc length s.
- Convert 60° to radians: θ = 60 × π/180 = π/3 ≈ 1.0472 rad.
- Write the arc length formula: s = r × θ.
- Substitute the values: s = 5 × 1.0472.
- Compute the result: s ≈ 5.236 m. This is one-sixth of the full circumference (2π × 5 ≈ 31.42 m).
The arc is exactly one-sixth of the circumference because 60° is one-sixth of 360°.
When to Use Each Variable
- Solve for Arc Length — when you know the radius and central angle, e.g., calculating the length of curved track on a racetrack.
- Solve for Radius — when you know the arc length and angle, e.g., determining the curve radius from a measured road arc.
- Solve for Angle — when you know the arc length and radius, e.g., finding the central angle subtended by a measured arc on a wheel.
Key Concepts
The arc length formula s = rθ is one of the simplest and most fundamental relationships in geometry. It connects the linear distance along a curve to the radius and central angle measured in radians. A radian is defined so that the arc length equals the radius when θ = 1, which is what makes the formula so elegant and unit-free.
Applications
- Road design: calculating the length of highway curves and interchange ramps
- Manufacturing: determining belt lengths for pulleys and circular conveyors
- Astronomy: converting angular separation between stars to physical distance at a known range
- Architecture: computing material lengths for arched windows, doorways, and domes
Common Mistakes
- Forgetting to convert degrees to radians — the formula s = rθ requires θ in radians, not degrees
- Confusing arc length with chord length — the arc follows the curve while the chord is the straight-line distance between endpoints
- Using diameter instead of radius — the formula uses the radius r, which is half the diameter
Frequently Asked Questions
How do you calculate arc length?
Use the formula s = rθ, where r is the radius and θ is the central angle in radians. If the angle is in degrees, convert first by multiplying by π/180. For example, a 10 m radius circle with a 90° angle has arc length s = 10 × (90π/180) = 10 × 1.5708 ≈ 15.71 m.
What is the formula for arc length?
The arc length formula is s = rθ (radians) or equivalently s = πrθ/180 (degrees). Here s is the arc length, r is the radius, and θ is the central angle. The formula follows directly from the definition of a radian.
How do you find the length of an arc without knowing the angle?
You need at least two of the three variables (s, r, θ) to find the third. Without the angle, you can determine it if you know the arc length and radius: θ = s/r. If you only know the chord length and radius, you can find the angle using θ = 2 × arcsin(chord / 2r), then compute the arc length.
What is the difference between arc length and chord length?
Arc length is the distance measured along the curved path of the circle between two points, while chord length is the straight-line distance between those same two points. The arc is always longer than the chord (except for a zero-degree angle where both are zero). For small angles, they are nearly equal.
Why must the angle be in radians for s = rθ?
A radian is defined as the angle where the arc length equals the radius (s = r when θ = 1 rad). This definition makes the formula s = rθ exact with no conversion factor. Using degrees requires an extra factor of π/180.
What is the arc length of a semicircle?
A semicircle has a central angle of π radians (180°). Its arc length is s = rπ, which is half the circumference. For r = 10, the semicircular arc is 10π ≈ 31.42.
How is arc length related to circumference?
The full circumference is the arc length for a complete 360° (2π radian) rotation: C = 2πr. Any arc is a fraction of this, proportional to the central angle: s = rθ. A 90° arc is one-quarter of the circumference, a 180° arc is half, and so on.
Arc Length Formula
The arc length formula connects the distance along a curve to the circle's radius and the central angle:
s = r × θ
Where:
- s — arc length (same unit as the radius)
- r — radius of the circle
- θ — central angle in radians (multiply degrees by π/180 to convert)
The formula works because a radian is defined so that the arc length equals the radius when θ = 1 rad. For a full circle (θ = 2π), the arc length equals the circumference 2πr.
Worked Examples
Road Design
What is the length of a highway curve with a 200 m radius and 45° turn?
A highway on-ramp curves through 45° with a centerline radius of 200 m. Find the curve length.
- Convert angle: 45° × π/180 = 0.7854 rad
- s = 200 × 0.7854
- s ≈ 157.08 m
Road designers use this to calculate pavement length, guardrail runs, and lane markings along curved sections.
Architecture
How much trim molding is needed for a semicircular arched doorway?
An arched doorway has a radius of 0.5 m (semicircular arch, 180°). How long is the trim piece?
- Convert angle: 180° × π/180 = π rad
- s = 0.5 × π
- s ≈ 1.5708 m
Builders measure the arc to cut flexible trim, crown molding, or decorative bands that follow the curve.
Manufacturing
What length of pipe stock is needed for a 90° bend with a 0.3 m bend radius?
A plumber bends a copper pipe through 90° around a mandrel with a 0.3 m radius. How much pipe does the bend consume?
- Convert angle: 90° × π/180 = π/2 rad
- s = 0.3 × 1.5708
- s ≈ 0.4712 m
Knowing the arc length prevents cutting pipe too short and accounts for material consumed by the bend.
Related Calculators
- Circle Calculator — Calculate area, circumference, radius, and diameter.
- Circle Sector Calculator — Find sector area from radius and central angle.
- Circle Segment Calculator — Compute chord length, segment height, and segment area.
- Circular Motion Calculator — Apply arc geometry to physics rotation problems.
- Length Converter — Convert between meters, feet, inches, and other length units.
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