Math & Conversions Arc Length Calculator Arc Length Calculator ($L$) Calculate the length of the arc ($L$) along the circumference of a circle, determined by its radius ($r$) and the central angle ($\theta$). Enter your values below and select whether the angle is in degrees or radians. Enter Dimensions Radius ($r$) Central Angle ($\theta$) Degrees ($^\circ$) Radians Calculate Arc Length Key Results Arc Length ($L$) Circumference ($C$) Sector Area ($\mathcal{A}$) Step-by-Step Solution Formulas for Arc Length The arc length is defined as a fraction of the circle's total circumference. The formula used depends on whether the central angle ($\theta$) is measured in radians or degrees. 1. Arc Length using Radians ($\theta$) The simplest and most fundamental relationship: the arc length ($L$) is the product of the radius and the angle (in radians). $$L = r \theta$$ This is the preferred formula for scientific and engineering calculations because of its simplicity. 2. Arc Length using Degrees ($\theta^\circ$) The formula for degrees explicitly shows the ratio of the partial angle to the full circle: $$L = 2\pi r \left( \frac{\theta}{360^\circ} \right)$$ Radians Conversion If you start with degrees, you must first convert the angle to radians to use the simpler formula: $$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180^\circ}$$ Arc Length vs. Chord Length It is important to distinguish the arc length from the chord length (which is the straight-line distance between the two endpoints of the arc). The arc length is always greater than the chord length, except when the angle is $0^\circ$ or $360^\circ$, where they are equal. Frequently Asked Questions (FAQ) What is the formula for arc length in radians? When the central angle ($\theta$) is measured in radians, the arc length ($L$) is calculated by the formula $L = r \\theta$. What is the formula for arc length in degrees? When the central angle ($\theta$) is measured in degrees, the arc length ($L$) is calculated as a fraction of the total circumference: $L = 2\pi r \left( \\frac{\\theta}{360^\\circ} \\right)$. What is the difference between arc length and chord length? The **arc length** is the measurement of the curved path along the circumference between the two endpoints of the radii. The **chord length** is the straight-line distance connecting those same two endpoints. How does this differ from the arc length formula in Calculus? In Calculus, the arc length formula is used to find the length of a curve defined by a function $f(x)$ over an interval $[a, b]$, using an integral: $L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$. This geometric calculator only addresses the arc length of a perfect circle. Arc Length Key Formulas Length (Radians) $$L = r \theta$$ Length (Degrees) $$L = C \frac{\theta}{360^\circ}$$ Sector Area $$\mathcal{A} = \frac{1}{2} r L$$ Related Circle Tools Sector Area Calculator Circle Calculator (Area, Circumference) Central Angle Calculator Degrees to Radians Converter Chord Length Calculator

Subcategories in Math & Conversions Arc Length Calculator Arc Length Calculator ($L$) Calculate the length of the arc ($L$) along the circumference of a circle, determined by its radius ($r$) and the central angle ($\theta$). Enter your values below and select whether the angle is in degrees or radians. Enter Dimensions Radius ($r$) Central Angle ($\theta$) Degrees ($^\circ$) Radians Calculate Arc Length Key Results Arc Length ($L$) Circumference ($C$) Sector Area ($\mathcal{A}$) Step-by-Step Solution Formulas for Arc Length The arc length is defined as a fraction of the circle's total circumference. The formula used depends on whether the central angle ($\theta$) is measured in radians or degrees. 1. Arc Length using Radians ($\theta$) The simplest and most fundamental relationship: the arc length ($L$) is the product of the radius and the angle (in radians). $$L = r \theta$$ This is the preferred formula for scientific and engineering calculations because of its simplicity. 2. Arc Length using Degrees ($\theta^\circ$) The formula for degrees explicitly shows the ratio of the partial angle to the full circle: $$L = 2\pi r \left( \frac{\theta}{360^\circ} \right)$$ Radians Conversion If you start with degrees, you must first convert the angle to radians to use the simpler formula: $$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180^\circ}$$ Arc Length vs. Chord Length It is important to distinguish the arc length from the chord length (which is the straight-line distance between the two endpoints of the arc). The arc length is always greater than the chord length, except when the angle is $0^\circ$ or $360^\circ$, where they are equal. Frequently Asked Questions (FAQ) What is the formula for arc length in radians? When the central angle ($\theta$) is measured in radians, the arc length ($L$) is calculated by the formula $L = r \\theta$. What is the formula for arc length in degrees? When the central angle ($\theta$) is measured in degrees, the arc length ($L$) is calculated as a fraction of the total circumference: $L = 2\pi r \left( \\frac{\\theta}{360^\\circ} \\right)$. What is the difference between arc length and chord length? The **arc length** is the measurement of the curved path along the circumference between the two endpoints of the radii. The **chord length** is the straight-line distance connecting those same two endpoints. How does this differ from the arc length formula in Calculus? In Calculus, the arc length formula is used to find the length of a curve defined by a function $f(x)$ over an interval $[a, b]$, using an integral: $L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$. This geometric calculator only addresses the arc length of a perfect circle. Arc Length Key Formulas Length (Radians) $$L = r \theta$$ Length (Degrees) $$L = C \frac{\theta}{360^\circ}$$ Sector Area $$\mathcal{A} = \frac{1}{2} r L$$ Related Circle Tools Sector Area Calculator Circle Calculator (Area, Circumference) Central Angle Calculator Degrees to Radians Converter Chord Length Calculator.