1. What is the Inscribed Angle Theorem?

The Inscribed Angle Theorem states that an angle inscribed in a circle is half the measure of its intercepted arc (or the central angle that subtends the same arc). This fundamental theorem in geometry relates inscribed angles to central angles in a circle.

2. How Does the Calculator Work?

The calculator uses the Inscribed Angle Theorem formula:

Where:

• Inscribed Angle — Angle formed by two chords in a circle which have a common endpoint
• Central Angle — Angle whose vertex is the center of the circle and whose sides (rays) extend to the endpoints of the arc

Explanation: The calculator can compute either the inscribed angle from a known central angle, or vice versa, by applying the theorem in both directions.

3. Importance of the Theorem

Details: This theorem is crucial in circle geometry, helping to solve problems involving cyclic quadrilaterals, arc measures, and angle relationships in circles. It's widely used in geometric proofs and practical applications like architecture and engineering.

4. Using the Calculator

Tips: Select whether you want to calculate the inscribed or central angle, then enter the known angle value in degrees (must be between 0 and 360).

5. Frequently Asked Questions (FAQ)

Q1: Does the theorem work for any inscribed angle?
A: Yes, as long as both sides of the angle are chords of the circle and the vertex is on the circumference.

Q2: What if the central angle is more than 180 degrees?
A: The theorem still applies, but the inscribed angle will be greater than 90 degrees (obtuse).

Q3: How is this related to Thales' theorem?
A: Thales' theorem is a special case where the central angle is 180°, making the inscribed angle 90°.

Q4: Can this be used for angles in radians?
A: Yes, the relationship holds for radians as well, though this calculator uses degrees.

Q5: Does the circle's size affect the relationship?
A: No, the theorem is independent of the circle's radius - it depends only on angle measures.