What is the relationship between inscribed and central angles?

An inscribed angle is always half the measure of the central angle that subtends the same arc. If the central angle is θ, the inscribed angle is θ/2.

How do I find the angle between a tangent and a chord?

The angle between a tangent and a chord equals the inscribed angle on the opposite arc. This is known as the tangent-chord angle theorem.

What is Thales' theorem?

Thales' theorem states that any angle inscribed in a semicircle is a right angle (90°). This is a special case of the inscribed angle theorem.

Circle Theorems Calculator

Calculate circle geometry theorems: inscribed angles, central angles, and tangent-chord angles. This calculator provides a mini solver for various circle theorem relationships.

Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)

Circle Theorems Calculator

Theorem Type

Central Angle (degrees)

How It Works

Inscribed Angle Theorem: An inscribed angle is half the measure of its intercepted arc (or the central angle subtending the same arc).

Central Angle Theorem: A central angle is twice the measure of any inscribed angle subtending the same arc.

Tangent-Chord Angle Theorem: The angle between a tangent and a chord equals the inscribed angle on the opposite arc.

All angles subtending the same arc are equal (inscribed angles in the same segment).

The angle in a semicircle is always 90° (Thales' theorem).

Step-by-step Examples

Example 1: Inscribed Angle from Central Angle

Given a central angle of 80°, find the corresponding inscribed angle.

Given: Central angle = 80°
Apply theorem: Inscribed angle = Central angle / 2
Calculate: Inscribed angle = 80° / 2 = 40°
The inscribed angle subtending the same arc is half the central angle.

Example 2: Central Angle from Inscribed Angle

Given an inscribed angle of 30°, find the corresponding central angle.

Given: Inscribed angle = 30°
Apply theorem: Central angle = 2 × Inscribed angle
Calculate: Central angle = 2 × 30° = 60°
The central angle is twice the inscribed angle subtending the same arc.

Common Mistakes

Forgetting to divide by 2 when finding inscribed angle from central angle.
Confusing inscribed angle with central angle - remember inscribed = central / 2.
Not recognizing that angles subtending the same arc are related.
Mixing up the tangent-chord angle with the inscribed angle on the opposite arc.

Use Cases

Mathematics: Solving geometry problems involving circle theorems.
Education: Teaching students about circle geometry and angle relationships.
Engineering: Analyzing circular components and angle measurements.
Architecture: Designing circular structures with specific angle requirements.
Surveying: Calculating angles in circular measurements and boundaries.

Frequently Asked Questions

Common questions about circle theorems.

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