Inscribed Angle Theorem Calculator
Inscribed angle = ½ × Central angle = ½ × Arc
Calculate From
Inscribed Angle Theorem
- • Inscribed angle = ½ × intercepted arc
- • Inscribed angle = ½ × central angle
- • Central angle = arc measure
Circle Diagram
Understanding the Inscribed Angle Theorem
The Inscribed Angle Theorem states that an inscribed angle is half of the central angle that subtends the same arc. This is one of the most important theorems in circle geometry.
Key Definitions
- Inscribed Angle: An angle with vertex on the circle and sides as chords
- Central Angle: An angle with vertex at the center of the circle
- Intercepted Arc: The arc cut off by an angle's sides
- Arc Measure: The degree measure of an arc, equal to its central angle
Important Corollaries
Inscribed in Semicircle
An angle inscribed in a semicircle is always 90° (the central angle is 180°).
Same Arc Theorem
Inscribed angles that intercept the same arc are equal.
The Formula
Inscribed Angle = ½ × Central Angle = ½ × Arc
Where all angles are measured in degrees
Frequently Asked Questions
What is an inscribed angle?
An inscribed angle has its vertex on the circle and its sides are chords. It "intercepts" or "cuts off" an arc of the circle.
Why is the inscribed angle half the central angle?
This can be proven using the isosceles triangle formed by two radii and a chord, along with the exterior angle theorem. The math shows the inscribed angle is exactly half.
What is the inscribed angle in a semicircle?
An inscribed angle in a semicircle (intercepting the diameter) is always 90°. This is because the central angle is 180°, and half of 180° is 90°.
Can inscribed angles share the same arc?
Yes! All inscribed angles intercepting the same arc are equal, regardless of where the vertex is on the circle. This is the "Inscribed Angles Subtending the Same Arc" theorem.
What is arc measure vs arc length?
Arc measure is the angle in degrees. Arc length is the actual distance along the curve: length = (arc°/360°) × 2πr.
What is a reflex arc?
A reflex arc is the "other" arc - if an arc is 90°, its reflex is 360° - 90° = 270°. It's the major arc when the original is the minor arc.