Central Angle Calculator
Calculate central angle in radians and degrees from arc length and radius. Free online central angle calculator with step-by-step results and formula explanations for geometry and trigonometry problems.
About This Calculator
About Central Angle
A central angle is an angle formed at the center of a circle by two radii that intersect the circumference at two distinct points. The arc between those two points is the intercepted arc, and the central angle determines its length. Understanding central angles is fundamental to circle geometry, trigonometry, and countless real-world applications.
Central Angle Formula
The central angle theta in radians is given by the simple ratio theta = s / r, where s is the arc length and r is the circle's radius. To express the angle in degrees, multiply by the conversion factor: theta deg = (s / r) x (180 / pi). This formula works because the definition of a radian is the angle that subtends an arc equal in length to the radius.
Relationship Between Central Angle, Arc Length, and Radius
For any circle, the central angle is directly proportional to the arc length and inversely proportional to the radius. Doubling the arc length while keeping the radius constant doubles the central angle. Doubling the radius while keeping the arc length constant halves the central angle. This proportional relationship is why the formula theta = s / r is so elegant.
Radians vs Degrees
Radians are the natural unit for measuring angles in mathematics because they simplify formulas. One radian is the angle that produces an arc length equal to the radius. A full circle is 2pi radians (about 6.283 rad) or 360 degrees. Degrees divide a full circle into 360 equal parts, which is convenient for everyday measurements. The calculator displays results in both units automatically.
Real-World Applications
Central angle calculations appear in navigation (computing great-circle distances on Earth), astronomy (measuring angular diameters of celestial bodies), engineering (designing circular segments for gears, arches, and curved structures), computer graphics (circular motion and rotation), and physics (angular velocity and rotational motion).
How to Use This Calculator
Enter the arc length and radius in any consistent unit of length (both must use the same unit). Click Calculate to see the central angle in both radians (accurate to 4 decimal places) and degrees (accurate to 2 decimal places). All inputs are saved to the URL for easy sharing -- you can bookmark or share the link with the results pre-filled.
Common Mistakes to Avoid
The most common error is using different units for arc length and radius -- they must match (e.g., both in meters or both in feet). Another is entering the diameter instead of the radius. Always use the radius (half the diameter) for the formula theta = s / r to work correctly.
Frequently Asked Questions
How do you calculate the central angle from arc length and radius?
The central angle in radians is calculated by dividing the arc length by the radius: theta = s / r. To convert to degrees, multiply the result by 180/pi. For example, if the arc length is 10 units and the radius is 5 units, the central angle is 2 radians or approximately 114.59 degrees.
What is the formula for central angle in degrees?
When the arc length and radius are known, the central angle in degrees is theta = (s / r) x (180 / pi), where s is the arc length and r is the radius. This formula first finds the angle in radians (s/r) and then converts to degrees by multiplying by 180/pi.
What is a central angle in geometry?
A central angle is an angle whose vertex is at the center of a circle and whose sides (rays) extend to intersect the circle at two points. The arc between those two points is called the intercepted arc, and the central angle determines what fraction of the full circle the arc represents.
How do you find the arc length from the central angle?
Arc length is found using the formula s = r x theta, where r is the radius and theta is the central angle in radians. If the angle is in degrees, convert to radians first by multiplying by pi/180, then apply the formula. This is the inverse operation of finding the central angle from arc length.
What is the relationship between radians and degrees for central angles?
A full circle equals 2pi radians (approximately 6.283 rad) or 360 degrees. One radian equals about 57.3 degrees. To convert radians to degrees, multiply by 180/pi. To convert degrees to radians, multiply by pi/180. This calculator provides both units for every central angle result.
What is the maximum possible central angle?
A central angle can be any non-negative value. For a single sector within a circle, the central angle is typically between 0 deg and 360 deg (0 to 2pi radians). However, central angles can exceed 360 deg when considering multiple rotations -- for example, an angle of 720 deg represents two full rotations around the circle.
What units should I use for arc length and radius?
Arc length and radius can be in any unit of length (meters, centimeters, feet, inches, etc.), as long as both values use the same unit. The central angle result is a dimensionless ratio (radians) and its degree equivalent, so it does not depend on the length unit chosen.