Central Angle Calculator
Calculate the central angle of a circle given the arc length and radius using theta = arc/radius. The central angle is the angle at the center subtended by an arc, fundamental to circle geometry, surveying, navigation, and engineering applications.
This free online central angle calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Minimum: 0.01
Length of the arc along the circle
Minimum: 0.01
Radius of the circle
Results
Central Angle (degrees)
114.5916 degrees
Central Angle (radians)
2
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Central Angle Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Central Angle Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Central Angle Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
Related Calculators
Chord Length Calculator
Calculate the length of a chord in a circle given the radius and the central angle subtended by the chord. Uses the formula c = 2r*sin(theta/2), essential for geometry, engineering, surveying, and arc-based constructions in architecture and design.
Circular Segment Area Calculator
Calculate the area of a circular segment (the region between a chord and its arc) given the radius and central angle. Uses A = (r^2/2)(theta - sin(theta)), essential for engineering, architecture, tank volume calculations, and curved surface analysis.
Arc Length Calculator
Calculate the length of a circular arc given the radius and central angle in degrees.
Circle Area Calculator
Calculate the area of a circle from its radius using the formula A = πr². Essential for geometry, engineering, and everyday measurements.
Circumscribed Circle (Circumradius) Calculator
Calculate the circumradius (radius of the circumscribed circle) of a triangle given its three sides using R = abc/(4*Area). The circumscribed circle passes through all three vertices. Used in geometry, trigonometry, navigation, and satellite positioning.
Inscribed Angle Calculator
Calculate the inscribed angle from the central angle using the inscribed angle theorem: inscribed = central/2. An inscribed angle has its vertex on the circle and is always half the central angle subtending the same arc, fundamental to circle theorems.
About Central Angle Calculator
The Central Angle Calculator determines the angle at the center of a circle that corresponds to a given arc length and radius. The relationship theta = s/r (where s is arc length and r is radius) defines the radian measure and is one of the most fundamental formulas in mathematics. This formula connects linear and angular measurements: knowing any two of {arc length, radius, central angle} lets you find the third. Central angles are used in surveying (measuring angular spans), navigation (bearing calculations), engineering (gear tooth spacing), and astronomy (angular diameter of celestial objects). The calculator outputs the angle in both degrees and radians.
The Math Behind It
Formula Reference
Central Angle from Arc Length
theta (radians) = arc length / radius
Variables: arc length = curved distance along the circle, r = radius
Worked Examples
Example 1: Arc 10, Radius 5
Find the central angle for arc length 10 and radius 5.
The central angle is 2 radians (114.59 degrees).
Example 2: Quarter Circle
A circle has radius 8. Find the central angle for an arc of length 4*pi.
The central angle is pi/2 radians (90 degrees), a quarter circle.
Common Mistakes & Tips
- !Confusing arc length with chord length. Arc length follows the curve of the circle; chord length is the straight-line shortcut.
- !Forgetting to convert between radians and degrees. The formula theta = s/r gives the answer in radians. Multiply by 180/pi for degrees.
- !Using diameter instead of radius. The formula uses the radius r, not the diameter d. If given the diameter, divide by 2.
Related Concepts
Used in These Calculators
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Frequently Asked Questions
What is a radian?
A radian is the angle subtended by an arc whose length equals the radius. Since the circumference is 2*pi*r, a full circle is 2*pi radians. One radian is approximately 57.3 degrees. Radians are the natural unit for angles in calculus and physics.
How does the central angle relate to the inscribed angle?
The inscribed angle theorem states that an inscribed angle is exactly half the central angle that subtends the same arc. If the central angle is 80 degrees, any inscribed angle on the same arc is 40 degrees.
Can the central angle be greater than 360 degrees?
The central angle for a single arc is between 0 and 360 degrees. However, if the arc length exceeds the circumference (wraps around more than once), the formula gives theta > 2*pi, representing more than one full revolution.
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