⭕ Central Angle Calculator
Calculate the central angle of a circle from radius and arc length
Enter the radius of the circle
Enter the length of the arc
How to Use This Calculator
Enter Radius
Input the radius of the circle. This is the distance from the center to any point on the circle.
Enter Arc Length
Input the length of the arc. This is the distance along the curved portion of the circle.
Calculate
Click "Calculate Central Angle" to find the angle in degrees and radians, along with chord length and sector area.
Formula
θ = s / r
Where θ is in radians. To convert to degrees: θ (degrees) = θ (radians) × 180° / π
Where:
- θ = Central angle (in radians)
- s = Arc length
- r = Radius of the circle
Related Formulas:
- Chord Length = 2r × sin(θ/2)
- Sector Area = (θ × r²) / 2
- Arc Length = r × θ (when θ is in radians)
Example 1: Find the central angle for a circle with radius 5 and arc length 10
θ = s / r = 10 / 5 = 2 radians
θ (degrees) = 2 × 180° / π ≈ 114.59°
Example 2: Find the central angle for a circle with radius 8 and arc length 12
θ = s / r = 12 / 8 = 1.5 radians
θ (degrees) = 1.5 × 180° / π ≈ 85.94°
About Central Angle Calculator
The Central Angle Calculator finds the angle at the center of a circle that subtends (forms) a given arc. The central angle is directly related to the arc length and radius through the simple formula θ = s/r, where θ is in radians.
When to Use This Calculator
- Geometry: Calculate angles in circular segments
- Engineering: Design curved structures and arcs
- Architecture: Plan circular features and arches
- Navigation: Calculate angles for circular routes
- Physics: Analyze rotational motion and angular displacement
- Surveying: Measure angles in circular plots
Why Use Our Calculator?
- ✅ Simple Input: Just radius and arc length
- ✅ Multiple Results: Angle, chord length, and sector area
- ✅ Degrees & Radians: Results in both units
- ✅ Instant Calculation: Get answers immediately
- ✅ 100% Accurate: Precise mathematical formulas
- ✅ Completely Free: No registration required
Understanding Central Angles
A central angle is an angle whose vertex is at the center of a circle and whose sides intersect the circle at two points, creating an arc. Key relationships:
- Arc Length: Directly proportional to the central angle. Larger angle = longer arc.
- Radius: Larger radius with same angle = longer arc length.
- Full Circle: 360° (2π radians) = circumference (2πr) arc length.
- Chord: The straight line connecting the two arc endpoints.
- Sector: The pie-shaped region bounded by the arc and two radii.
Real-World Applications
Clock Design: Calculate angles for hour markers. Each hour represents 30° (360° / 12).
Road Curves: Engineers use central angles to design curved road sections and calculate banking angles.
Architecture: Architects calculate central angles for designing arches, domes, and circular windows.
Frequently Asked Questions
What is a central angle?
A central angle is an angle whose vertex is at the center of a circle. Its sides intersect the circle at two points, creating an arc between them.
How do you calculate central angle from arc length?
Divide the arc length by the radius: θ = s/r. This gives the angle in radians. To convert to degrees, multiply by 180°/π.
What's the difference between central angle and inscribed angle?
A central angle has its vertex at the circle center. An inscribed angle has its vertex on the circle circumference. The inscribed angle is half the central angle subtending the same arc.
Can the central angle be greater than 360°?
In standard geometry, central angles are between 0° and 360° (0 to 2π radians). However, in some contexts (like rotations), angles can exceed 360°.
How do you find arc length from central angle?
Multiply the central angle (in radians) by the radius: s = r × θ. If the angle is in degrees, convert to radians first: s = r × (θ × π/180).
What is the chord length?
The chord length is the straight-line distance between the two endpoints of the arc. Formula: Chord = 2r × sin(θ/2), where θ is the central angle.
What is sector area?
Sector area is the area of the pie-shaped region bounded by the arc and two radii. Formula: Area = (θ × r²) / 2, where θ is in radians.
Why is the formula θ = s/r?
This comes from the definition of a radian: one radian is the angle that subtends an arc equal to the radius. So θ radians subtend an arc of length r × θ.