Free Inscribed Angle Calculator

The Inscribed Angle Calculator is a free online circle geometry tool that computes the angle formed when two chords intersect on a circle's circumference. Leveraging the inscribed angle theorem, this calculator can also serve as a central angle calculator and an arc length calculator, allowing users to obtain multiple circle parameters from a few inputs. Since the term “chord angle” is a synonym for inscribed angle, this tool functions effectively as a chord angle calculator as well.

What Is an Inscribed Angle?

In circle geometry, an inscribed angle ($\theta_i$) has its vertex on the circumference and its sides formed by two chords. The central angle ($\theta_c$) shares the same chord endpoints but is centered at the interior of the circle. The connection between these two angles is the foundation of many circular properties.

The Inscribed Angle Theorem

The inscribed angle theorem states that an inscribed angle is exactly half the measure of the central angle that subtends the same arc. Mathematically:

$$\theta_i = \frac{\theta_c}{2}$$

A key consequence is that all inscribed angles intercepting the same arc are equal, no matter where the vertex lies on the circle. A well‑known special case is the angle in a semicircle: when the chord is a diameter, the central angle is 180° and the inscribed angle becomes 90° (a right angle).

Central Angle and Arc Length

The arc length $L$ that corresponds to a central angle $\theta_c$ (in radians) is:

$$L = r \cdot \theta_c$$

where $r$ is the radius. Equivalently, the central angle (in radians) can be expressed as:

$$\theta_c = \frac{L}{r}$$

Using the inscribed angle theorem ($\theta_c = 2\theta_i$), the arc length can be computed directly from the inscribed angle:

$$L = r \cdot 2\theta_i$$

This formula requires $\theta_i$ in radians. To convert between degrees and radians, use:

$$\theta_{\text{rad}} = \frac{\pi}{180^\circ} \cdot \theta_{\text{deg}} \qquad \theta_{\text{deg}} = \frac{180^\circ}{\pi} \cdot \theta_{\text{rad}}$$

Worked Examples

Example 1: Central angle → Inscribed angle
If $\theta_c = 120^\circ$, then:

$$\theta_i = \frac{120^\circ}{2} = 60^\circ$$

Example 2: Inscribed angle → Arc length
A circle has radius 2 m and $\theta_i = 45^\circ$.

1. Convert to radians: $45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}$ rad.
2. Compute arc length: $L = 2 \times 2 \times \frac{\pi}{4} = \pi \approx 3.14$ m.

Example 3: Arc length → Central and inscribed angles
Given $L = 5$ m and $r = 2$ m:

1. $\theta_c = \frac{5}{2} = 2.5$ rad $\approx 143.24^\circ$.
2. $\theta_i = \frac{2.5}{2} = 1.25$ rad $\approx 71.62^\circ$.

These examples illustrate how the inscribed angle theorem streamlines conversions between angles and arc length.

How to Use the Calculator

Input(s)	Output(s)
Central angle (degrees or radians)	Inscribed angle
Radius and arc length	Central angle and inscribed angle
Radius and inscribed angle	Arc length
Radius and central angle	Arc length (and inscribed angle)

The tool accepts angles in either degrees or radians and performs the necessary conversions automatically.

Practical Applications

The inscribed angle theorem is widely applied in navigation (calculating bearings), architectural design (shaping arches), and solving complex circle geometry problems. Having a reliable inscribed angle calculator simplifies these tasks, making it useful for students and professionals alike.

Summary

Thanks to the inscribed angle theorem, circle geometry tasks that involve chords, arcs, and angles become straightforward. This Inscribed Angle Calculator consolidates the roles of a central angle calculator, arc length calculator, and chord angle calculator into one versatile tool, delivering fast and accurate results.