Central Angle of a Circle Formula

What is a Central Angle?

To explain the Central angle definition, the central angle is formed when two radii intersect at the centre of a circle. Keep in mind that a vertex is a point at which two lines intersect to form an angle. The vertex of a central angle is always the centre point of a circle.

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Central Angle Theorem

Before we can understand the central angle theorem, we must first understand what subtended and inscribed angles are, as they are mentioned in the definition. A subtended angle is an angle formed by an object at a specific outer position.


If that definition doesn't make sense to you, consider this: You're standing on Earth, looking up at the sun. A three-sided triangle is formed. The three sides are the sun ray that travels from the top of the sun to your eyes, the sun ray that travels from the bottom of the sun to your eyes, and the sun's height.


According to the Central Angle Theorem, the central angle subtended by two points on a circle is always twice the inscribed angle subtended by those points.


How to Find a Central Angle with Arc Length?

The formula to measure central angle (which is measured in radians). 

Central angle formula equals - (Length of the arc)/(Length of the radius).

(arc length) ÷ circumference equals (central angle) ÷ 360°

The formula of circumference = 2πr


Central Angle Formula = ( Arc Length * 360°) / 2πr


The central angle will be expressed in degrees. When you think about it, this formula makes sense. The length of the arc in relation to the total length around the circle (circumference) is the same proportion as the angle of the arc in relation to the total angle in a circle (360 degrees).

How to Find the Central Angle from the Arc Length and Radius?

You can also use the radius of the circle as well as the arc length to find the central angle. Call the measure of the central angle which is denoted by the symbol θ. Then:

θ = s ÷ r, where s is the arc length as well as r is the radius. θ is measured in radians.

Again, depending on the information available, you can rearrange this equation. The radius and central angle can be used to calculate the length of the arc. You can also find the radius if you know the central angle and the length of the arc.

If you want the arc length, then the discussed equation looks like this:

s = θ * r, where s is the arc length, r is the radius, and θ is the central angle which is measured in radians.

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FAQs (Frequently Asked Questions)

1) Define the Measure of the Central Angle of a Circle.

Answer) A central angle has an apex (vertex) that is the centre O of a circle and legs (sides) that are radii that intersect the circle in two distinct points. The arc's angular distance is also known as the central angle.The size of a central angle Θ is 0° < Θ < 360° or 0 < Θ < 2π (radians).

2) What is the Central Angle Equal To?

Answer) The arc length divided by the radius of the circle is the measure of an arc. In radians, the arc measure equals the corresponding central angle measure. That is why radians are so natural: a central angle of one radian spans an arc that is exactly one radius long.

3) Is 180 a Major Arc?

Answer) A minor arc is less than 180 degrees in length. A major arc is defined as an arc with a length greater than 180 degrees. Because it divides the circle in half, an arc with a measure of 180 degrees is known as a semicircle. The vertex of a central angle is generally the centre of a circle.