What are Central Angles?
Central angles are angles that measure the amount of rotation from the center of a circle. The vertex of the angle must be at the center of the circle and the endpoints of the angle must be on the circumference of the circle.
Central Angles Describe Arcs
An arc is a portion of the circumference of a circle. We often use central angles to describe arcs because the central angle determines how much of the circumference the arc covers.
A \(360^\circ\) angle is a full rotation. So, the full circumference of a circle could be described as an arc with a measure of \(360^\circ\). You could also describe it as an arc of \(2\pi\) radians because \(2\pi\) radians is equivalent to \(360^\circ\).
A \(180^\circ\) angles is a half rotation and the diameter of a circle splits the circle and the circumference exactly in half. So, the diameter of any circle will always form a \(180^\circ\) central angle.
Central Angle Formula
You can calculate the central angle with this formula if you know the length of the arc and the radius of the circle. Just be aware that the formula gives you the measurement of the angle in radians. If you need to know the measurement in degrees, then you can convert the radian measurement to degrees.
\(\theta =\) Central Angle (in radians)
\(S =\) Arc Length
\(r =\) Radius
Inscribed Angle Theorem
The inscribed angle theorem says that if a central angle and an inscribed angle subtend the same arc, then the measurement of the central angle will be twice as big as the measurement of the inscribed angle.
When Will I Use Central Angles?
When you calculate the arc length of a circle, you will use the central angle to determine how much of the total circumference the arc covers.
When you find the area of a sector, you will use the central angle to determine how much area the sector covers compared to the entire circle.
You can use the inscribed angle theorem to find the measurements of central angles if you know the measurements of the inscribed angles that subtend the same arcs.