Central Angle of Circle: Formula and Solved examples

The central angle of a circle formula calculates the angle between two radii of a circle. A central angle can also be defined as an angle subtended by the arc of a circle at the center of the circle. Thus, the vertex of the central angle will always be the center point of the circle. The two points of the circle, where the radii intersect in the circle, forms a segment of the circle called the Arc length.

Key Terms: Circle, Angle Of Circle, Central Angle Theorem, Central Angle of a Circle Formula, Arc, Radius

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Central Angle Theorem

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Before jumping on central angle theorem, let us 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.

For example, let's consider that 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.

Central Angle Theorem

Central angle theorem states that the central angle subtended by two points on a circle is twice the inscribed angle subtended by those points.

If ∠AOB is the central angle and ∠ACB is the inscribed angle of the circle then,

∠AOB = 2∠ACB

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Central Angle of a Circle Formula 

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To calculate the central angle using the central angle formula, we require the measure of the arc length that subtends the central angle at the center and the radius of the circle. The central angle of a circle formula is given by,

Central angle θ=Arc length ×360°2πr

Where, r is the radius of the circle. The central angle will be expressed in degrees.

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Central Angle Using Arc length and Radius 

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We can also use the radius of the circle and the arc length to find the central angle. The formula for that is given by,

Ѳ = s ÷ r

Where, s is the arc length and r is radius. Ѳ is measured in radians. Depending on the information given the radius and arc length can also be calculated by rearranging the equation.

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Things to Remember

• The length of the arc in relation to the total length around the circle is the same proportion as the angle of the arc in relation to the total angle in a circle (360°).
• The central angle is also known as the arc’s angular distance.
• If the intersection points A and B of the legs of the angle with the circle form a diameter, then Ѳ = 180° is a straight angle.
• An arc of a circle is a continuous portion of the circle. It consists of two endpoints and all the points on the circle between these endpoints.
• An inscribed angle is the angle formed between two chords when they meet on the boundary of the circle.
• The angle in a semicircle is always 90°.

Also Read: Areas Related to Circles Formula