Imagine you are standing at the center of a perfectly circular arena, looking out at the crowds around the edge. You spot a friend in one seat and a teacher in another. How far apart are they? What angle do the lines of sight between you and each of them form? These questions are answered by the concept of a central angle.
A central angle is the angle formed by two radii drawn from the center of the circle to two points on its circumference. The two radii are the lines of sight from you to the friend and to the teacher. The angle between them is the central angle, the angle closest to the center of the circle.
The friend and teacher sit on the circumference of the circle. The curved path along the edge that connects them is called an arc.
If you know the arc length (the distance you would walk along the arena to get from the friend to the teacher) and the total circumference of the circle, the relationship between the two is:
arc length / circumference = central angle / 360°
Rearranging gives:
central angle = (arc length / circumference) × 360°
This proportion works because the fraction of the circle’s perimeter that the arc occupies is exactly the same fraction of the full 360° angle.
When the radius r of the circle is known, you can compute the central angle in radians with:
θ = s / r
where s is the arc length. The result θ is measured in radians. If you prefer degrees, multiply the radian value by 57.2958 (or simply use the circumference method above).
You can also solve for the arc length:
s = θ × r
or for the radius when the arc length and central angle are given:
r = s / θ
Consider a third person—your neighbor—sitting on the opposite side of the arena. From the neighbor’s perspective, the two lines of sight to the friend and the teacher form an inscribed angle (an angle whose vertices lie on the circumference). The Central Angle Theorem links this inscribed angle to the central angle you observe:
∠AOC = 2 ∠ABC
Here, points A and B are the friend and teacher, C is the neighbor, and O is the center. The theorem holds when the neighbor lies on the same side of the chord AB as the arc that does not contain the other points.
When the inscribed point C moves inside the minor arc between A and B, the relationship changes. The inscribed angle becomes the supplement of half the central angle:
∠ABC = 180° – (∠AOC / 2)
In other words, the inscribed angle and half the central angle together sum to 180°.
Math Open Reference offers an interactive tool that lets you drag the neighbor around the circle and observe how the central and inscribed angles evolve in real time. Try it for a hands‑on understanding of the theory.
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