Central Angle in Degrees 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 determine the central angle in degrees when you know the arc length and radius of a circle segment.

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 circular segments and sectors.

Central angles are measured in degrees or radians. In this calculator, we focus on degrees, which are commonly used in everyday applications and geometric calculations.

How to Calculate Central Angle

To calculate the central angle in degrees, you need to know two key pieces of information:

The length of the arc (L)
The radius of the circle (r)

The central angle (θ) can be calculated using the formula:

Central Angle Formula

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

Formula

The central angle in degrees can be calculated using the following formula:

Central Angle Formula

θ = (L / r) × (180 / π)

This formula converts the ratio of arc length to radius (which is in radians) to degrees by multiplying by 180/π.

Worked Example

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

Example Calculation

Given:

Radius (r) = 5 units
Arc length (L) = 3 units

Using the formula:

θ = (3 / 5) × (180 / 3.14159) ≈ 103.6396 degrees

So, the central angle is approximately 103.64 degrees.

FAQ

What is the difference between a central angle and an inscribed angle?

A central angle is formed by two radii with a vertex at the center of the circle. An inscribed angle is formed by two chords with a vertex on the circumference. The central angle is always twice the inscribed angle that subtends the same arc.

How do I measure a central angle in a real-world scenario?

In practical applications, you can measure the arc length and radius using a ruler or measuring tape, then use this calculator to determine the central angle. For circular objects, you might use a protractor to measure the angle directly.

What units should I use for the radius and arc length?

The units for radius and arc length must be consistent. For example, if you measure the radius in centimeters, the arc length should also be in centimeters. The calculator will provide the angle in degrees regardless of the units used for length.