Subtended Angle — Definition, Formula & Examples

A subtended angle is the angle formed at a particular point when two lines (or rays) are drawn from that point to the endpoints of an arc or line segment. In circle geometry, an arc subtends an angle at the center or at any point on the circle.

Given an arc $AB$ of a circle with center $O$ , the angle subtended by the arc at $O$ is $\angle AOB$ . More generally, for any point $P$ on the major arc, the angle subtended by arc $AB$ at $P$ is $\angle APB$ , called an inscribed angle. The inscribed angle theorem states that the central angle is exactly twice any inscribed angle subtended by the same arc.

Key Formula

\theta_{\text{inscribed}} = \frac{1}{2}\,\theta_{\text{central}}

Where:

$\theta_{\text{inscribed}}$ = Angle subtended at a point on the circle (inscribed angle)
$\theta_{\text{central}}$ = Angle subtended at the center of the circle by the same arc

How It Works

To find a subtended angle, identify the arc or chord and the point where the angle is formed. If the point is the center of the circle, you get the central angle. If the point lies on the circle itself, you get an inscribed angle. The key relationship is that the central angle is always twice the inscribed angle when both are subtended by the same arc. This means every inscribed angle subtended by the same arc is equal, regardless of where the point sits on the major arc.

Worked Example

Problem: An arc AB subtends a central angle of 120° at the center O of a circle. Find the angle subtended by the same arc at a point P on the major arc.

Identify the relationship: The inscribed angle at P is half the central angle subtended by the same arc.

\angle APB = \frac{1}{2} \times \angle AOB

Substitute: Plug in the central angle of 120°.

\angle APB = \frac{1}{2} \times 120° = 60°

Answer: The angle subtended at point P on the major arc is 60°.

Why It Matters

Subtended angles appear throughout high school geometry proofs involving cyclic quadrilaterals, tangent-chord angles, and angles in semicircles. Understanding them is essential for GCSE and SAT geometry problems, and they form the basis of how surveyors and engineers measure angles from distant observation points.

Common Mistakes

Mistake: Using the wrong arc — calculating the inscribed angle from the minor arc when the point P lies on the minor arc.

Correction: The inscribed angle theorem applies when P is on the opposite arc from the one subtending the angle. Always check which arc the point P lies on and which arc defines the central angle.