The sagitta (also called the versine or rise) is the perpendicular height from the midpoint of a chord to the midpoint of the arc — it measures how much the arc 'bulges' above the chord. Formula: h = r(1 − cos(θ/2)) where r is the radius and θ is the central angle in radians. Given sagitta h and chord c, the radius is r = (4h² + c²)/(8h). This formula is essential in arch design, road surveying, and bridge engineering.

Arc Length Calculator — Geometry, Calculus & Parametric Curves

Geometry & Calculus

Arc Length Calculator

Q: How do you find arc length with radius and angle?

Use the arc length formula s = r × θ where θ must be in radians. Convert degrees to radians first: θ_rad = θ_deg × π/180. For example, for r = 5 and θ = 60°: θ_rad = 60 × π/180 = π/3 ≈ 1.0472, so s = 5 × π/3 ≈ 5.236. For a 90° arc (quarter circle) with r = 10: s = 10 × π/2 = 5π ≈ 15.708.

Q: What is the arc length formula in calculus?

The calculus arc length formula for y = f(x) from x = a to x = b is: s = ∫[a to b] √(1 + [f'(x)]²) dx. This comes from summing infinitesimal hypotenuses ds = √(dx² + dy²) = √(1 + (dy/dx)²) dx along the curve. For parametric curves x(t), y(t): s = ∫ √([dx/dt]² + [dy/dt]²) dt. For polar curves r = f(θ): s = ∫ √(r² + [dr/dθ]²) dθ.

Q: How do you find the radius from arc length?

There are four methods to find radius from arc length: (1) Given arc length s and central angle θ: r = s/θ (θ in radians). (2) Given chord c and angle θ: r = c/(2·sin(θ/2)). (3) Given sagitta h and chord c: r = (4h² + c²)/(8h) — the standard arch/road-design formula. (4) Radius of curvature of y = f(x) at point x: R = (1 + [f'(x)]²)^(3/2) / |f''(x)|.

Q: What is the difference between arc length and chord length?

Arc length is the distance measured along the curved path of the arc: s = rθ. Chord length is the straight-line distance between the two endpoints of the arc: c = 2r·sin(θ/2). The chord is always shorter than or equal to the arc length (c ≤ s), with equality only when the arc is a straight line (θ → 0). For a semicircle (θ = π) with r = 1: arc length = π ≈ 3.14159, chord length = 2. The difference grows larger for bigger angles.

Q: How do you calculate parametric arc length?

For parametric curves x = x(t), y = y(t) from t = t₁ to t = t₂, the arc length formula is s = ∫[t₁ to t₂] √([dx/dt]² + [dy/dt]²) dt. Example: for the unit circle x = cos(t), y = sin(t) from 0 to 2π: dx/dt = −sin(t), dy/dt = cos(t), integrand = √(sin²t + cos²t) = 1, so s = ∫₀²π 1 dt = 2π. For the cycloid x = t − sin(t), y = 1 − cos(t) from 0 to 2π: arc length = exactly 8.

Q: Why do most arc length integrals have no closed form?

The arc length integrand √(1 + [f'(x)]²) typically produces expressions that are not integrable in terms of elementary functions. For example, for y = x², f'(x) = 2x, and the integrand is √(1 + 4x²) — which requires hyperbolic functions or elliptic integrals for the antiderivative. Most functions, including sin(x), x^n for non-integer n, and ln(x), produce arc length integrands with no closed-form antiderivative. Numerical methods like Simpson's rule are used instead.

Calculate arc length using the geometry formula s = rθ, the calculus integral formula ∫√(1 + [f'(x)]²) dx, parametric curves, polar curves, and find radius from arc length using the sagitta-chord formula.

This arc length calculator covers four computational modes: circle arc geometry using the arc length formula s = rθ, calculus arc length via the integral formula ∫_a^b√(1 + [f'(x)]²) dx, parametric and polar arc length, and inverse radius-from-arc problems using the sagitta-chord formula r = (4h² + c²)/(8h). Every result includes full step-by-step working. For calculus modes, both symbolic (when available) and numerical (Simpson's rule, n = 1000) results are displayed.

Arc Length Calculator — Four Tools

Enter any two of the five quantities below. The calculator solves for all others using the arc length formula s = rθ. Leave unknown fields empty.

Angle unit for θ input:

r — Radius

θ — Central angle

s — Arc length

c — Chord length

h — Sagitta (arc height above chord)

r=5, θ=60°

r=10, s=15m

Semicircle r=1

h=2, c=10 (arch)

Full circle r=5

Error

— arc (s) - - chord (c) | sagitta (h)

Step-by-Step Working

Calculate the arc length of y = f(x) from x = a to x = b using the integral formula s = ∫_a^b√(1 + [f'(x)]²) dx. Symbolic integration is attempted first; numerical Simpson's rule (n = 1,000) is used as fallback and verification.

f(x) — function expression

Lower limit a

Upper limit b

Integration subintervals n: 1000

f(x)=x², [0,2]

f(x)=x², [0,1]

f(x)=sin(x), [0,π]

f(x)=x^(3/2), [0,4]

f(x)=x (line), [0,1]

Semicircle r=1

s = ∫_a^b √(1 + [f'(x)]²) dx Enter f(x), a, and b above — the formula will update

Error

Arc Length of f(x)

—

units

Step-by-Step Working

Arc length formula: s = ∫_t₁^t₂ √([dx/dt]² + [dy/dt]²) dt

x(t) expression

y(t) expression

t from (t₁)

t to (t₂)

Unit circle → s=2π

Ellipse 3×2

Cycloid → s=8

Arc length formula: s = ∫_α^β √(r² + [dr/dθ]²) dθ

r(θ) expression (use variable: theta)

θ from (α)

θ to (β)

Circle r=5 → s=10π

Archimedean spiral

Cardioid → s=8

Error

Parametric Arc Length

—

units

Step-by-Step Working

Formula: r = s / θ (θ in radians)

Arc length s

Central angle θ

Road: s=500m, θ=30°

s=10, θ=π (semicircle)

Sagitta-chord formula (arch / road design): r = (4h² + c²) / (8h)

r = (4h² + c²) / (8h) h = sagitta (rise of arc above chord) · c = chord length

Sagitta h (rise height)

Chord c

Bridge arch: h=5, c=30 → r=25

h=2, c=10 → r=6.5

Formula: r = c / (2·sin(θ/2))

Chord c

Central angle θ

c=5, θ=60° (equilateral → r=5)

Radius of curvature of y = f(x) at a given x: R = (1 + [f'(x)]²)^(3/2) / |f''(x)|

f(x) expression

x value

y=x² at x=1 → R≈5.59

y=sin(x) at x=0 → R=1

Error

Radius

—

Step-by-Step Working

Arc Length Formula — Geometry and Calculus

The arc length formula takes two forms depending on whether the curve is circular (constant curvature) or general (varying curvature). This arc length calculator handles both.

s = r × θ Geometry: circular arc · θ must be in radians · s = arc length · r = radius

s = ∫_a^b √(1 + [f'(x)]²) dx Calculus: any smooth curve y = f(x) · f'(x) = dy/dx

When to use each arc length formula

s = rθ (and also s = (θ/360) × 2πr in degrees): use when the curve is a circular arc with known radius r and central angle θ. This is the simple, direct formula for any arc that is part of a circle.
∫√(1 + [f'(x)]²) dx: use when the curve is any function y = f(x) — a parabola, sine curve, logarithm, etc. The arc length formula s = rθ does not apply to non-circular curves.
∫√([dx/dt]² + [dy/dt]²) dt: use for parametric curves x(t), y(t).
∫√(r² + [dr/dθ]²) dθ: use for polar curves r = f(θ).

Why θ must be in radians for s = rθ

The arc length formula s = rθ is derived from the definition of a radian: one radian is the angle subtended when arc length equals radius (s = r when θ = 1 rad). Therefore the formula s = r × θ works directly only in radians. To use degrees: first convert with θ_rad = θ_deg × π/180, then apply s = rθ. Equivalently in degrees: s = (θ/360) × 2πr = πrθ/180.

The most common arc length error: using θ in degrees directly in s = rθ. For r = 5 and θ = 60°, the correct answer is s = 5 × (60 × π/180) = 5 × π/3 ≈ 5.236, not s = 5 × 60 = 300. Always convert to radians first when using the arc length formula s = rθ.

How to Find Arc Length — Step-by-Step

Method 1: Geometry (Circular Arcs) — using s = rθ

Example 1: r = 5 cm, θ = 60°

Convert to radians: θ = 60 × π/180 = π/3 ≈ 1.0472 rad
Apply arc length formula s = rθ: s = 5 × π/3 = 5π/3 ≈ 5.236 cm
Chord: c = 2r·sin(θ/2) = 2×5×sin(30°) = 10×0.5 = 5.000 cm (equilateral triangle chord!)
Sagitta: h = r(1−cos(θ/2)) = 5(1−cos(30°)) = 5(1−0.866) ≈ 0.670 cm
Sector area: A = ½r²θ = ½×25×π/3 ≈ 13.09 cm²

Example 2: r = 10 m, s = 15 m → find θ

Rearrange arc length formula s = rθ: θ = s/r = 15/10 = 1.5 rad
Convert to degrees: 1.5 × 180/π ≈ 85.94°
Chord: c = 2×10×sin(1.5/2) = 20×sin(0.75) = 20×0.6816 ≈ 13.63 m

Example 3: Semicircle, r = 1, θ = 180°

θ = 180° = π rad. Apply arc length formula: s = 1 × π = π ≈ 3.14159 m
Chord = 2r·sin(π/2) = 2×1×1 = 2 m (diameter — the chord across a semicircle is the diameter)
Sagitta = r(1−cos(π/2)) = 1×(1−0) = 1 m = r (the arc rises by the full radius above the diameter chord)

Method 2: Calculus (Any Curve) — using ∫√(1 + [f'(x)]²) dx

Example 4: f(x) = x from 0 to 1 (straight line verification)

f'(x) = 1
Arc length integral: s = ∫₀¹ √(1 + 1²) dx = ∫₀¹ √2 dx = √2 × 1 = √2 ≈ 1.4142
Verification: Pythagorean theorem — line from (0,0) to (1,1) has length √(1² + 1²) = √2 ✓

Example 5: f(x) = x² from 0 to 2

f'(x) = 2x
Integrand: √(1 + 4x²)
No elementary closed form — compute numerically via Simpson's rule
s ≈ 4.6468 units

Example 6: Semicircle f(x) = √(1−x²) from −1 to 1

f'(x) = −x/√(1−x²)
Integrand: √(1 + x²/(1−x²)) = √(1/(1−x²)) = 1/√(1−x²)
s = ∫₋₁¹ 1/√(1−x²) dx = [arcsin(x)]₋₁¹ = π/2 − (−π/2) = π ≈ 3.14159 ✓

Arc Length Calculator — Calculus and the Integral Formula

The calculus arc length formula ∫√(1 + [f'(x)]²) dx is derived by treating the curve as a sequence of infinitesimal line segments. Each segment has horizontal width dx and vertical height dy = f'(x)dx. By the Pythagorean theorem, its length is ds = √(dx² + dy²) = √(1 + [f'(x)]²) dx. Integrating from a to b gives the total arc length.

s = ∫_a^b √(1 + [f'(x)]²) dx The calculus arc length formula — sums infinitesimal hypotenuses ds = √(dx² + dy²) along the curve

Why most arc length integrals have no closed form: The integrand √(1 + [f'(x)]²) typically produces expressions involving nested square roots or compositions that do not have antiderivatives in terms of elementary functions. For example, for y = x², the integrand √(1 + 4x²) requires the inverse hyperbolic sine function in its antiderivative. For y = sin(x), the integrand √(1 + cos²(x)) involves elliptic integrals. This arc length calculator uses Simpson's rule with 1,000 subintervals as the primary computation for functions without closed-form arc length integrals — achieving accuracy to approximately 8 significant figures for smooth functions.

The parametric extension of the arc length formula is: for x = x(t), y = y(t) from t₁ to t₂: s = ∫[t₁ to t₂] √([dx/dt]² + [dy/dt]²) dt. This follows the same infinitesimal argument: each element of arc has ds = √(dx² + dy²) = √([dx/dt]² + [dy/dt]²) dt.

Radius from Arc Length — Finding the Radius of a Curve

Finding the radius from arc length is the inverse problem: given measurements of the arc, recover the circle radius. There are four formulas for finding radius from arc length depending on what is known.

Formula 1: Radius from arc length and angle

Rearrange the arc length formula s = rθ: r = s / θ (θ in radians). This is the most direct way to find radius from arc length and angle. For example, a road curve with arc length s = 500 m and central angle θ = 30° = 0.5236 rad: r = 500/0.5236 ≈ 954.9 m.

r = s / θ Radius from arc length and central angle (θ in radians) · rearrangement of s = rθ

Formula 2: Sagitta-chord formula (arch and road design)

The most important formula for finding radius from arc length in engineering: given the sagitta h (the perpendicular height from chord midpoint to arc midpoint) and the chord c (straight-line distance between arc endpoints): r = (4h² + c²) / (8h)

r = (4h² + c²) / (8h) Sagitta-chord formula — arch design, road surveying, bridge engineering · h = sagitta · c = chord

Example: A bridge arch with chord c = 30 m and sagitta h = 5 m: r = (4×25 + 900)/(8×5) = (100 + 900)/40 = 1000/40 = 25 m. This sagitta-chord formula r = (4h² + c²)/(8h) is used universally in civil engineering for measuring the radius of road curves and bridge arches from field measurements.

Formula 3: Radius from chord and angle

From the chord formula c = 2r·sin(θ/2): r = c / (2·sin(θ/2))

Formula 4: Radius of curvature of y = f(x)

The osculating circle radius at a point x: R = (1 + [f'(x)]²)^(3/2) / |f''(x)|. For y = x² at x = 1: f'(1) = 2, f''(1) = 2, R = (1 + 4)^1.5/2 = 5^1.5/2 = 11.180/2 = 5.590.

How to find radius from arc length and angle: use r = s/θ with θ in radians. To find radius from arc length and chord only (no angle given), the equation 2r·sin(s/(2r)) = c must be solved numerically — this arc length calculator uses bisection to find r when both s and c are given.

Parametric and Polar Arc Length

Cycloid arc length (parametric, closed form)

For the cycloid x = t − sin(t), y = 1 − cos(t) from t = 0 to 2π: dx/dt = 1 − cos(t), dy/dt = sin(t). Integrand: √((1−cos t)² + sin²t) = √(2 − 2cos t). Using the identity 1 − cos(t) = 2sin²(t/2): integrand = 2|sin(t/2)|. Arc length = ∫₀²π 2sin(t/2) dt = [−4cos(t/2)]₀²π = −4(−1) + 4(1) = 8 exactly. This is one of the beautiful closed-form parametric arc lengths.

Cardioid arc length (polar, closed form)

For r = 1 + cos(θ) from θ = 0 to 2π: dr/dθ = −sin(θ). Integrand: √((1+cos θ)² + sin²θ) = √(2 + 2cos θ) = 2|cos(θ/2)|. Arc length = ∫₀²π 2|cos(θ/2)| dθ = 8 exactly. Two beautiful arc lengths that are both exactly 8 — the cycloid and the cardioid.

Common Mistakes in Arc Length Calculations

Mistake 1 — Using degrees in s = rθ (most common error)

❌ Wrong: s = 5 × 60 = 300 (θ = 60° used directly)
✅ Correct: convert first — θ = 60° × π/180 = π/3, then s = 5 × π/3 ≈ 5.236

Mistake 2 — Forgetting the +1 under the radical in the calculus formula

❌ Wrong: s = ∫√([f'(x)]²) dx = ∫|f'(x)| dx (this is not arc length)
✅ Correct: s = ∫√(1 + [f'(x)]²) dx — the 1 represents the horizontal component dx in the Pythagorean theorem

Mistake 3 — Confusing chord length with arc length

Chord c = 2r·sin(θ/2) is always shorter than arc length s = rθ (for θ > 0)
For a 60° arc with r = 5: arc length ≈ 5.236 vs chord = 5.000 — visually similar but numerically different
For a 180° arc: arc length = πr ≈ 3.14r vs chord = 2r — the arc is 57% longer than the chord

Mistake 4 — Using s = rθ for non-circular curves

❌ Wrong: applying arc length formula s = rθ to a parabola, sine curve, or ellipse
✅ Correct: the arc length formula s = rθ applies only to circular arcs. Use the integral formula for any non-circular curve.

Mistake 5 — Integrating in the wrong direction (b < a)

The arc length integral ∫ₐᵇ√(1+[f'(x)]²)dx gives a negative value if b < a, but arc length is always positive
Always integrate from the smaller limit to the larger: if a > b, swap them, since arc length is the same either way

Worked Examples — Arc Length Calculator in Practice

Problem 1: Full circle circumference check (r = 5, θ = 360°)

θ = 360° = 2π rad
Arc length s = rθ = 5 × 2π = 10π ≈ 31.416 (this is the circumference — full circle arc)
Verification: C = 2πr = 2π×5 = 10π ✓

Problem 2: Find arc length of f(x) = x^(3/2) from 0 to 4

f'(x) = (3/2)x^(1/2) = (3/2)√x
Integrand: √(1 + (9/4)x)
This has a closed form: antiderivative involves (1 + 9x/4)^(3/2)
s = [(8/27)×(1+9x/4)^(3/2)]₀⁴ = (8/27)×(10)^(3/2) − (8/27)×1 = (8/27)(31.623−1) ≈ 9.073

Problem 3: Arch design — find radius from sagitta and chord

Given: arch with chord c = 10 m, sagitta h = 2 m
Apply sagitta-chord formula r = (4h² + c²)/(8h) = (4×4 + 100)/(8×2) = (16 + 100)/16 = 116/16 = 7.25 m
Find arc length: cos(θ/2) = 1 − h/r = 1 − 2/7.25 = 0.7241 → θ/2 = 43.62° → θ = 87.24° = 1.523 rad
Arc length s = rθ = 7.25 × 1.523 ≈ 11.04 m

Problem 4: Parametric arc length — ellipse

x = 3cos(t), y = 2sin(t), t from 0 to 2π
dx/dt = −3sin(t), dy/dt = 2cos(t)
Integrand: √(9sin²t + 4cos²t) — no elementary antiderivative (elliptic integral)
Numerical: s ≈ 15.865

Problem 5: Polar arc — circle r = 5 full revolution

r = 5 (constant), dr/dθ = 0
Integrand: √(25 + 0) = 5
s = ∫₀²π 5 dθ = 5×2π = 10π ≈ 31.416 ✓

Problem 6: Find arc length — sin(x) from 0 to π

f'(x) = cos(x), integrand: √(1 + cos²x)
No closed form — involves elliptic integrals
Numerical (Simpson, n=1000): s ≈ 3.8202

Problem 7: Road curve radius from arc and angle

Arc length s = 500 m, central angle θ = 30° = 0.5236 rad
Radius from arc length: r = s/θ = 500/0.5236 ≈ 954.9 m
Chord: c = 2×954.9×sin(15°) = 1909.8×0.2588 ≈ 494.3 m

Problem 8: Radius of curvature of y = sin(x) at x = π/2

f'(π/2) = cos(π/2) = 0, f''(π/2) = −sin(π/2) = −1
R = (1 + 0)^(3/2) / |−1| = 1/1 = 1
At the peak of sin(x), the osculating circle has radius 1 — the sine wave has the same curvature as a unit circle at its maximum

Common Questions About Arc Length

What is the arc length formula?

For a circular arc with radius r and central angle θ (in radians), the arc length formula is s = r × θ. In degrees: s = (θ/360) × 2πr. For a curve y = f(x) from x = a to x = b, the calculus arc length formula is s = ∫[a to b] √(1 + [f'(x)]²) dx. The geometry formula s = rθ applies only to circular arcs; the integral applies to any smooth curve. This arc length calculator handles both.

How do you find arc length with radius and angle?

Use arc length = r × θ where θ must be in radians. Convert degrees first: θ_rad = θ_deg × π/180. Example: r = 5, θ = 60° → θ_rad = π/3 → arc length s = 5 × π/3 ≈ 5.236. The formula s = rθ is only valid in radians — using degrees directly in s = rθ is the most common arc length error.

What is the arc length formula in calculus?

The calculus arc length formula for y = f(x) from a to b is s = ∫[a to b] √(1 + [f'(x)]²) dx. It comes from summing infinitesimal hypotenuses ds = √(dx² + dy²) along the curve. For parametric x(t), y(t): s = ∫√([dx/dt]² + [dy/dt]²) dt. For polar r = f(θ): s = ∫√(r² + [dr/dθ]²) dθ. The integral formula ∫√(1 + [f'(x)]²) dx must appear instead of simply ∫f'(x)dx — the +1 represents the horizontal path element.

How do you find the radius from arc length?

Four methods: (1) Given s and θ: r = s/θ (θ in radians). (2) Given sagitta h and chord c: r = (4h² + c²)/(8h) — the standard arch formula. (3) Given chord c and angle θ: r = c/(2·sin(θ/2)). (4) Radius of curvature of f(x) at x: R = (1 + [f'(x)]²)^(3/2)/|f''(x)|. The sagitta-chord formula r = (4h² + c²)/(8h) is the most practical for field measurements in engineering and surveying.

What is the difference between arc length and chord length?

Arc length s = rθ is the distance measured along the curved path. Chord length c = 2r·sin(θ/2) is the straight-line distance between the two arc endpoints. The chord is always shorter than or equal to the arc (c ≤ s). For a semicircle with r = 1: arc = π ≈ 3.14, chord = 2 (diameter). The sagitta h = r(1 − cos(θ/2)) measures how far the arc bulges above the chord — all three quantities describe the arc differently.

What is a sagitta?

The sagitta (h) is the perpendicular distance from the midpoint of the chord to the midpoint of the arc — it measures how much the arc "bulges" above the chord. Formula: h = r(1 − cos(θ/2)). Given sagitta and chord: r = (4h² + c²)/(8h). The sagitta-chord formula r = (4h² + c²)/(8h) is essential in bridge engineering, road design, and rail surveying.

How do you calculate parametric arc length?

For parametric curves x(t), y(t) from t₁ to t₂: arc length = ∫ √([dx/dt]² + [dy/dt]²) dt. For the unit circle x = cos(t), y = sin(t), t from 0 to 2π: integrand = √(sin²t + cos²t) = 1, so arc length = 2π. For the cycloid x = t−sin(t), y = 1−cos(t): arc length = exactly 8. Most parametric arc lengths require numerical integration.

Why do most arc length integrals have no closed form?

The integrand √(1 + [f'(x)]²) typically creates expressions not integrable in elementary functions. For f(x) = x², the integrand √(1 + 4x²) requires inverse hyperbolic functions. For f(x) = sin(x), the integrand involves elliptic integrals. This is why this arc length calculator uses Simpson's rule (n = 1000) for numerical arc length — achieving ~8 significant figures of accuracy for smooth functions.

Related Calculators

Arc Length Formulas

Circle arc (rad)s = rθ

Circle arc (deg)s = πrθ/180

Chordc = 2r·sin(θ/2)

Sagittah = r(1−cosθ/2)

Sector areaA = ½r²θ

Calculus f(x)∫√(1+f'²)dx

Parametric∫√(ẋ²+ẏ²)dt

Polar∫√(r²+r'²)dθ

Radius from Arc

From s and θr = s/θ

From h and c(4h²+c²)/8h

From c and θc/(2sin(θ/2))

Curvature(1+f'²)^1.5/|f''|

Known Arc Lengths

Circumference2πr

Semicircleπr

Quarter circleπr/2

Cycloid (1 arch)8r

Cardioid8a

Line y=x, [0,1]√2

Arc Length Calculator — Geometry, Calculus & Parametric Curves

Arc Length Calculator

Arc Length Formula — Geometry and Calculus

When to use each arc length formula

Why θ must be in radians for s = rθ

How to Find Arc Length — Step-by-Step

Method 1: Geometry (Circular Arcs) — using s = rθ

Example 1: r = 5 cm, θ = 60°

Example 2: r = 10 m, s = 15 m → find θ

Example 3: Semicircle, r = 1, θ = 180°

Method 2: Calculus (Any Curve) — using ∫√(1 + [f'(x)]²) dx

Example 4: f(x) = x from 0 to 1 (straight line verification)

Example 5: f(x) = x² from 0 to 2

Example 6: Semicircle f(x) = √(1−x²) from −1 to 1

Arc Length Calculator — Calculus and the Integral Formula

Radius from Arc Length — Finding the Radius of a Curve

Formula 1: Radius from arc length and angle

Formula 2: Sagitta-chord formula (arch and road design)

Formula 3: Radius from chord and angle

Formula 4: Radius of curvature of y = f(x)

Parametric and Polar Arc Length

Cycloid arc length (parametric, closed form)

Cardioid arc length (polar, closed form)

Common Mistakes in Arc Length Calculations

Mistake 1 — Using degrees in s = rθ (most common error)

Mistake 2 — Forgetting the +1 under the radical in the calculus formula

Mistake 3 — Confusing chord length with arc length

Mistake 4 — Using s = rθ for non-circular curves

Mistake 5 — Integrating in the wrong direction (b < a)

Worked Examples — Arc Length Calculator in Practice

Problem 1: Full circle circumference check (r = 5, θ = 360°)

Problem 2: Find arc length of f(x) = x^(3/2) from 0 to 4

Problem 3: Arch design — find radius from sagitta and chord

Problem 4: Parametric arc length — ellipse

Problem 5: Polar arc — circle r = 5 full revolution

Problem 6: Find arc length — sin(x) from 0 to π

Problem 7: Road curve radius from arc and angle

Problem 8: Radius of curvature of y = sin(x) at x = π/2

Common Questions About Arc Length

Related Calculators

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