Calculate Circular Arc

Online calculator and formula for calculating a circular arc

Circular Arc Calculator

Calculate Circular Arc

Calculates the arc length and the central angle of a circular arc from radius and given parameter.

Radius of the circle
Depending on calculation mode
Choose Calculation Mode
Determines which parameter is used as input
Results
Arc length:
Angle in degrees:

Visualization

M r α l

A circular arc is a part of the circle's circumference, bounded by two radii.
The central angle α determines the length of the arc l.

What is a Circular Arc?

A circular arc is a part of the circle's circumference:

  • Circle segment: Section of the circle line between two points
  • Central angle: Angle between two radii from the center
  • Arc length: Distance along the circle line
  • Proportionality: Length proportional to the angle
  • Radius dependent: Larger radius = longer arc
  • Applications: Architecture, mechanical engineering, navigation

The Radius as Fundamental Parameter

The radius is the fundamental parameter for all circle calculations:

Definition

Distance from the center to any point on the circle line. Constant for all points of the circle.

Influence on Arc Length

Doubled radius leads to doubled arc length at the same angle. Linear relationship.

Central Angle and Radian Measure

The central angle can be expressed in different units:

Degrees (°)
\[360° = \text{Full circle}\]

Common unit, 360 parts for full circle

Radians (rad)
\[2\pi \text{ rad} = \text{Full circle}\]

Mathematical unit, based on π

Circumference and Proportionality

The circular arc is a proportional part of the circle's circumference:

Circumference
\[U = 2\pi r\]

Total circumference of the circle

Proportion
\[\frac{\alpha}{360°} = \frac{l}{2\pi r}\]

Ratio of angle to total angle

Circular Arc Formulas

Basic Formula - Arc Length from Angle (Degrees)
\[l = \frac{2\pi r \cdot \alpha}{360°}\]

Arc length proportional to central angle in degrees

Angle from Arc Length
\[\alpha = \frac{360° \cdot l}{2\pi r}\]

Central angle in degrees

Radian Measure
\[l = r \cdot \alpha_{rad}\]

Simple formula in radian measure

Conversion Degrees ↔ Radians
\[\alpha_{rad} = \frac{\pi \cdot \alpha_{deg}}{180°}\]

Conversion between angle units

Circular Sector Area
\[A = \frac{\pi r^2 \cdot \alpha}{360°}\]

Area of the corresponding circular sector

Symbols and Notations
  • l: Arc length
  • r: Radius of the circle
  • α: Central angle
  • π: Pi (≈ 3.14159)
  • °: Degree
  • rad: Radian

Example

Example Calculation
r = 50 α = 50°
1. Calculate arc length
\[l = \frac{2\pi \cdot 50 \cdot 50°}{360°}\]
\[l = \frac{5000\pi}{360°} \approx 43.63\]

The arc length is approximately 43.63 units

2. Convert to radians
\[\alpha_{rad} = \frac{\pi \cdot 50°}{180°} \approx 0.873\]

50° corresponds to approximately 0.873 radians

3. Check with radian measure
\[l = r \cdot \alpha_{rad} = 50 \cdot 0.873 \approx 43.63\]

Confirms the result

Practical Applications
  • Architecture: Arch bridges, vaults
  • Mechanical engineering: Gears, cams
  • Navigation: Course calculations

The Circular Arc in Mathematics and Practice

A circular arc is a fundamental geometric object that describes a section of the circle line between two points. It is uniquely determined by the central angle α and the radius r and plays an important role in mathematics, engineering and architecture.

Mathematical Definition

A circular arc is the part of a circle line that is bounded by two radii. The most important parameters are:

  • Radius r: Distance from the center to the circle line
  • Central angle α: Angle between the two bounding radii
  • Arc length l: Length of the arc along the circle line

Fundamental Relationships

\[l = \frac{2\pi r \cdot \alpha}{360°} \quad \text{(Angle in degrees)}\] \[l = r \cdot \alpha \quad \text{(Angle in radians)}\]

These formulas show the direct proportionality between arc length, radius and angle.

Understanding Angle Units

Degrees (°)

The degree measure divides the full circle into 360 equal parts. It is the most intuitive unit for everyday applications.

Practical Angles
  • 90°: Quarter circle
  • 180°: Semicircle
  • 270°: Three-quarter circle
Radians (rad)

The radian measure is the natural angle unit of mathematics. One radian is the angle where the arc length equals the radius.

Important Values
  • π/2 rad: 90° (Quarter circle)
  • π rad: 180° (Semicircle)
  • 2π rad: 360° (Full circle)

Applications in Practice

Circular arcs are found everywhere in our daily life and in technology:

  • Architecture: Arch bridges, vaults, domes, window arches
  • Mechanical engineering: Gears, cams, cam mechanisms
  • Road construction: Curve radii, roundabouts
  • Navigation: Course calculations, great circles on the globe
  • Optics: Lenses, mirrors, prisms
  • Sports: Running tracks, field markings

Related Concepts

Circular Sector

The circular sector is the area between two radii and the circular arc. Its area is A = (πr²α)/360°.

Circular Segment

The circular segment is the area between the circular arc and the chord. It is smaller than the circular sector.

Chord

The chord is the straight connection between the endpoints of the circular arc. It is always shorter than the arc.

Secant

A secant is a straight line that intersects the circle at two points. The chord is a part of the secant.

Calculation Strategies

Depending on the given parameters, there are different calculation approaches:

  1. Given: Radius and angle → Calculate arc length
  2. Given: Radius and arc length → Calculate angle
  3. Given: Angle and arc length → Calculate radius

Historical Significance

The calculation of circular arcs belongs to the oldest mathematical problems. Already the ancient Greeks and Babylonians dealt with circle geometry. The development of trigonometry was closely linked to the need to calculate circular arcs precisely.

Modern Applications

In modern technology, circular arc calculations are indispensable:

  • CAD software: Construction of curves and arcs
  • CNC programming: Path planning for machine tools
  • Robotics: Motion planning along curved paths
  • Computer animation: Interpolation between keyframes
  • GPS navigation: Calculation of shortest paths on the globe

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