Central angle and inscribed angle

Given an arc $AB$, we can observe it from two points of view: from the centre $O$ of the circle, or from a point $P$ on the circle itself. The link between the two angles is surprisingly simple.

Theorem — Central angle and inscribed angle

All the inscribed angles subtending the same arc are congruent to one another, and each is half of the corresponding central angle: $\alpha_{\text{circ}} = \frac{\alpha_{\text{centro}}}{2} = \frac{\text{arco}}{2\cdot\text{raggio}} \quad\text{(in radianti)}$

In the figure, the central angle $\widehat{AOB}$ is $2\alpha$, while the inscribed angles $\widehat{APB}$ and $\widehat{AQB}$, which subtend the same arc $AB$, are both equal to $\alpha$.

The central angle $\widehat{AOB}=2\alpha$ is twice each inscribed angle $\widehat{APB}=\widehat{AQB}=\alpha$ that subtends the same arc.

In the following simulation you can check the theorem yourself: drag the points $A$, $B$ and $P$ along the circle and observe that the central angle always stays twice the inscribed angle.

Drag $A$, $B$ and $P$ along the circle: the central angle $\widehat{AOB}$ is always twice the inscribed angle $\widehat{APB}$ (as long as $P$ stays on the major arc).

Links

Topics: Euclidean circle
Concepts: Central angle · Inscribed angle · Arc
Skills: Synthetic geometry