Find The Measure in Degrees of The Central Angle Calculator

A central angle is an angle whose vertex is at the center of a circle and whose sides (rays) extend to the circumference. This calculator helps you find the measure of a central angle in degrees when you know the arc length and radius of the circle.

What is a Central Angle?

A central angle is an angle whose vertex is at the center of a circle and whose sides extend to the circumference. Central angles are fundamental in geometry and are used to calculate various properties of circles, including arc lengths and sector areas.

Central angles are measured in degrees or radians. This calculator focuses on degree measurements, which are more commonly used in practical applications.

How to Calculate Central Angles

To calculate a central angle, you need to know either:

The arc length and the radius of the circle
The sector area and the radius of the circle

The most common scenario is calculating the angle when you know the arc length and radius. The formula for this calculation is:

θ = (L / r) × (180 / π)

Where:

θ is the central angle in degrees
L is the arc length
r is the radius of the circle
π (pi) is approximately 3.14159

The Formula

The formula for calculating a central angle in degrees is derived from the relationship between arc length, radius, and angle in a circle. Here's the step-by-step breakdown:

Divide the arc length (L) by the radius (r) to get the angle in radians: (L / r)
Convert radians to degrees by multiplying by (180 / π)

θ (degrees) = (L / r) × (180 / π)

This formula works because a full circle is 360 degrees (or 2π radians), so the conversion factor (180/π) properly scales the radian measure to degrees.

Worked Example

Let's calculate the central angle for a circle with radius 5 meters and arc length 3 meters.

Divide the arc length by the radius: 3 / 5 = 0.6 radians
Convert radians to degrees: 0.6 × (180 / 3.14159) ≈ 34.377 degrees

The central angle is approximately 34.38 degrees.

Note: The calculator uses π ≈ 3.14159 for these calculations. For more precise results, you can use a more accurate value of π if needed.

FAQ

What is the difference between a central angle and an inscribed angle?: A central angle has its vertex at the center of the circle, while an inscribed angle has its vertex on the circumference. The central angle is always twice the inscribed angle that subtends the same arc.
Can I calculate a central angle if I know the sector area?: Yes, you can use the formula θ = (A / (r²)) × (180 / π), where A is the sector area and r is the radius. This formula works because the area of a sector is (θ/360) × πr².
What units should I use for the radius and arc length?: The units for radius and arc length must be consistent (e.g., both in meters or both in inches). The calculator will return the angle in degrees regardless of the units used for length.
Is there a maximum angle I can calculate with this tool?: The calculator can handle any angle from 0 to 360 degrees. Angles greater than 360 degrees would represent more than one full rotation around the circle.
How accurate are the calculations?: The calculator uses π ≈ 3.14159 for its calculations, which provides accurate results for most practical purposes. For applications requiring higher precision, you may need to use more decimal places for π.