1. Central and Inscribed Angle Relationship

In circle geometry, a central angle is an angle whose vertex is at the center of the circle, while an inscribed angle has its vertex on the circle. When both angles subtend the same arc, the central angle is always twice the measure of the inscribed angle.

2. How the Calculator Works

The calculator uses the fundamental relationship:

You can calculate either:

• The central angle when you know the inscribed angle
• The inscribed angle when you know the central angle

3. Geometric Principles

Key Theorem: The Central Angle Theorem states that the central angle subtended by two points on a circle is twice any inscribed angle subtended by those same points.

Visualization: If you draw lines from the center and from a point on the circumference to the same two points on the circle, this relationship becomes apparent.

4. Using the Calculator

Steps:

1. Select whether you're calculating from central or inscribed angle
2. Enter the known angle value in degrees
3. Click "Calculate" to get the result

5. Frequently Asked Questions (FAQ)

Q1: Does this relationship hold for all circles?
A: Yes, this is a fundamental property of Euclidean geometry that applies to all circles.

Q2: What if the angles subtend different arcs?
A: The relationship only holds when both angles subtend exactly the same arc of the circle.

Q3: Can I use radians instead of degrees?
A: Yes, the relationship works the same in radians since it's a proportional relationship.

Q4: What about angles outside the circle?
A: This calculator specifically deals with central and inscribed angles. Other angle relationships (like angles formed by tangents) have different properties.

Q5: Why is this relationship important?
A: It's fundamental for solving many geometric problems involving circles and is widely used in architecture, engineering, and design.