Arc Length Calculator
Calculate the length of a curve $L = \int_a^b \sqrt{1+[f'(x)]^2} \, dx$
$$ L = \int_{0}^{4} \sqrt{1 + \left[ \frac{d}{dx}\left( x^{3/2} \right) \right]^2} \, dx $$
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Arc Length
Visualized Arc Segment
Step-by-Step Solution
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By Prof. David Anderson
Professor of Mathematics | 20+ Years Experience
"The problem with 'Arc Length' is that it means two different things. For an engineer building an archway, it's about finding the curve length using Chord and Height. For a calculus student, it's about solving the Arc Length Integral. I designed this Arc Length Calculator to handle both: whether you are cutting metal or solving for the length of a function curve on an exam."
The Ultimate Guide to Arc Length: Geometry, Calculus, Parametric, and Polar Curves
How to Use an Arc Length Calculator for Circles, Functions, and Engineering
Arc Length is the total distance along a curved line. Finding this value is a fundamental task that bridges standard Geometry and advanced Calculus. The method you choose to calculate arc length depends entirely on the data you have.
If you are working with circle sectors (like pizza slices, gears, or arches), you need the Geometric Arc Length Formula which involves radius and angle. If you are analyzing a graph or a projectile path, you need the Calculus Arc Length Formula which uses derivatives and integration. Our versatile Arc Length Calculator handles both modes instantly to give you precise results.
1. Geometry Mode: Arc of a Circle
When calculating the arc length of a circle, the formula changes slightly depending on whether your central angle ($\theta$) is measured in degrees or radians.
Circular Arc Formulas
$$ s = r \theta \quad \text{(Radians)} $$
$$ s = 2\pi r \left( \frac{\theta}{360} \right) \quad \text{(Degrees)} $$
$$ s = 2\pi r \left( \frac{\theta}{360} \right) \quad \text{(Degrees)} $$
For Engineers: Chord & Height
In real-world scenarios like sheet metal bending or woodworking, you often don't know the radius. Instead, you can measure the Chord Length ($c$) and the Arch Height ($h$) (sagitta). Our calculator solves this by first finding the radius:
$r = \frac{c^2}{8h} + \frac{h}{2}$, then using it to calculate the arc length.
2. Calculus Mode: Deriving the Formula
How does an Arc Length Calculator work for functions? It applies the Pythagorean Theorem to infinitely small sections of the curve.
Imagine zooming into a curve until it looks straight. This tiny hypotenuse ($ds$) has legs $dx$ and $dy$.
1. $ds^2 = dx^2 + dy^2$
2. Factor out $dx^2$: $ds^2 = [1 + (\frac{dy}{dx})^2] dx^2$
3. Take the square root and sum them up (integrate) to find the total Curve Length:
Calculus Arc Length Formula
$$ L = \int_{a}^{b} \sqrt{1 + \left[ f'(x) \right]^2} \, dx $$
3. Advanced: Parametric and Polar Curves
In Calculus II, you often need to find the arc length of curves that aren't simple functions $y=f(x)$.
Type A
Parametric Curves $(x(t), y(t))$
If the position is defined by time $t$, the Parametric Arc Length formula integrates the speed:
$$ L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt $$
Type B
Polar Curves $r(\theta)$
For spirals and cardioids defined by $r(\theta)$, use the Polar Arc Length Formula:
$$ L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta $$
4. How to Calculate Arc Length (Step-by-Step)
Follow this rigorous protocol to solve arc length calculus problems manually using the integral formula.
Step 1
Find the Derivative f'(x)
First, differentiate the function $y = f(x)$.
Example: If $y = x^2$, then $f'(x) = 2x$.
Step 2
Square and Add 1
Prepare the integrand. Square the derivative and add 1.
$\sqrt{1 + (2x)^2} = \sqrt{1 + 4x^2}$
Step 3
Integrate
Set up the definite integral with bounds $a$ and $b$ and solve. (Note: Many arc length integrals require U-Substitution or Trig Substitution).
5. Master Class: Examples
Type A: Circular Arc
Geometry
Find the arc length of a circle with radius 10 cm and central angle $45^\circ$.
$s = 2\pi(10) \times \frac{45}{360}$
$s = 20\pi \times \frac{1}{8} = 2.5\pi \approx 7.85 \text{ cm}$.
$s = 20\pi \times \frac{1}{8} = 2.5\pi \approx 7.85 \text{ cm}$.
Type B: Curve Length
Calculus Integration
Find the length of $y = \frac{2}{3}x^{3/2}$ from $x=0$ to $x=1$.
1. Derive: $y' = x^{1/2} = \sqrt{x}$.
2. Setup: $\sqrt{1 + (\sqrt{x})^2} = \sqrt{1+x}$.
3. Integrate: $\int_0^1 \sqrt{1+x} \, dx$.
Use U-Sub ($u=1+x$): $[\frac{2}{3}(1+x)^{3/2}]_0^1$.
Result: $\frac{2}{3}(2^{3/2} - 1^{3/2}) \approx 1.22$.
6. Professor's FAQ
Radians vs Degrees?
Always check your angle units! The simple formula $s = r\theta$ ONLY works if $\theta$ is in radians. If you have degrees, you must convert ($\times \pi/180$) or use the degree-specific formula.
How do I find Arc Length with just Chord and Height?
This is common in construction. Use our calculator's "Chord & Height" mode. It internally calculates the radius $r$ first ($r = \frac{c^2}{8h} + \frac{h}{2}$), then the angle, and finally the arc length.
Can arc length be negative?
No. Arc length is a physical distance, so it must be positive. If your integral gives a negative result, you likely mixed up the bounds ($a$ and $b$) or made an algebra error in your calculation.
What if the integral is too hard to solve?
Calculus arc length integrals often result in messy square roots that cannot be solved algebraically (e.g., Ellipses). In these cases, we use Numerical Integration (like Simpson's Rule) to approximate the value.
References & Further Reading
- Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage Learning. (Section 8.1: Arc Length).
- Math Open Reference. "Arc Length of a Circle."
- Paul's Online Math Notes. "Arc Length." Lamar University.
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