NUM8ERS Geometry Tools ⌒ Circles & Curves Step-by-Step Solutions

Arc Length Calculator

Q: What is the formula for arc length?

For a circle: Arc Length = (θ/360°) × 2πr (degrees) or Arc Length = θ × r (radians), where θ is the central angle and r is the radius.

Q: How do I convert degrees to radians?

To convert degrees to radians, multiply by π/180. For example: 90° = 90 × (π/180) = π/2 radians. To convert radians to degrees, multiply by 180/π.

Q: What is a sector of a circle?

A sector is a 'pie slice' of a circle bounded by two radii and an arc. The arc length is the curved edge of the sector. Sector area = (θ/360°) × πr² (degrees) or (1/2)θr² (radians).

Q: How is arc length used in real life?

Arc length is used in road design (curves), roller coaster engineering, calculating belt/chain lengths, architecture (arches), clock hands, satellite orbits, and manufacturing curved surfaces.

Q: What is the arc length formula in calculus?

For a curve y = f(x) from x = a to x = b: Arc Length = ∫[a to b] √(1 + (dy/dx)²) dx. For parametric curves: L = ∫ √((dx/dt)² + (dy/dt)²) dt.

Calculate arc length for circles, sectors, and curves. Supports degrees and radians with step-by-step solutions and additional sector calculations.

🧮 Calculator 📚 All Formulas 🔄 Conversions ❓ FAQ

🧮 Arc Length Calculator

Select a calculation mode, enter your values, and get instant results with step-by-step solutions.

⌒

Arc Length from Radius & Angle

Formula

s = (θ/360°) × 2πr

📖 Definition

Arc length is the distance along the curved line of an arc. It is a portion of the circumference, determined by the central angle and radius.

Angle Unit:

Length Unit:

Radius (r)

Central Angle (θ) in degrees

Arc Length (s)

s = (90/360) × 2π × 5 = 7.854

Circumference

—

Arc %

—

Chord Length

—

◔

Sector Arc Length & Area

Formulas

Arc = θr | Area = ½θr²

📖 Definition

A sector is a "pie slice" of a circle. This calculator finds both the arc length (curved edge) and the area of the sector.

Angle Unit:

Radius (r)

Central Angle (θ) in degrees

Arc Length

Sector Area

—

Perimeter

—

% of Circle

—

⌓

Arc Length from Chord & Height

Formula

s = 2r × arcsin(c/2r)

📖 Definition

When you know the chord length (straight distance) and sagitta (height of arc), you can calculate the arc length and radius.

Chord Length (c)

Sagitta / Arc Height (h)

Arc Length

Radius

—

Central Angle

—

Arc Height

—

📚 Arc Length Formulas

Complete reference for all arc length and related formulas.

⌒ Arc Length (Degrees)

s = (θ/360) × 2πr

θ = central angle in degrees, r = radius. Calculates the curved distance.

⌒ Arc Length (Radians)

s = θ × r

θ = central angle in radians, r = radius. Simpler formula when using radians.

◔ Sector Area

A = (θ/360) × πr²

Area of a "pie slice" sector. Also: A = ½θr² in radians.

⌓ Chord Length

c = 2r × sin(θ/2)

Straight-line distance between arc endpoints.

↕ Sagitta (Arc Height)

h = r(1 - cos(θ/2))

Height from chord midpoint to arc. Also: h = r - √(r² - (c/2)²)

🔄 Find Radius from Chord

r = (c²/8h) + (h/2)

Calculate radius when you know chord length and sagitta.

🔄 Angle Conversions

Quick reference for converting between degrees and radians.

Common Angle Values

Degrees (°)	Radians	Exact Radians	% of Circle
30°	0.5236	π/6	8.33%
45°	0.7854	π/4	12.5%
60°	1.0472	π/3	16.67%
90°	1.5708	π/2	25%
120°	2.0944	2π/3	33.33%
180°	3.1416	π	50%
270°	4.7124	3π/2	75%
360°	6.2832	2π	100%

Conversion formulas: Degrees → Radians: multiply by π/180 | Radians → Degrees: multiply by 180/π

❓ Frequently Asked Questions

Common questions about arc length calculations answered.

What is arc length?+

Arc length is the distance along a curved line (arc) between two points. For a circle, it's the length of a portion of the circumference, determined by the central angle and radius.

What is the formula for arc length?+

For degrees: s = (θ/360°) × 2πr. For radians: s = θ × r. The radians formula is simpler because radians directly relate angle to arc length.

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

Arc length is the curved distance along the arc itself, while chord length is the straight-line distance between the two endpoints. Arc length is always ≥ chord length.

How do I convert degrees to radians?+

Multiply degrees by π/180. Example: 90° = 90 × (π/180) = π/2 ≈ 1.5708 radians. To convert radians to degrees, multiply by 180/π.

What is a sector of a circle?+

A sector is a "pie slice" of a circle bounded by two radii and an arc. The arc length is the curved edge. Sector area = (θ/360°) × πr² or (½)θr² in radians.

How is arc length used in real life?+

Arc length is used in road/railway curve design, roller coasters, belt/chain calculations, architecture (arches, domes), clock hands, satellite orbits, and manufacturing curved parts.

What is the arc length formula in calculus?+

For y = f(x) from a to b: L = ∫√(1 + (dy/dx)²) dx. For parametric: L = ∫√((dx/dt)² + (dy/dt)²) dt. These extend arc length to any curve.

Can arc length be negative?+

No, arc length is always positive or zero. It represents a physical distance along a curve, which cannot be negative regardless of direction.