What Is an Inscribed Angle?
An inscribed angle is an angle formed by two chords of a circle that share a common endpoint (the vertex) on the circle. The Inscribed Angle Theorem states that an inscribed angle is always exactly half of the central angle that subtends the same arc. This calculator instantly converts a central angle into its corresponding inscribed angle.
How to Use the Calculator
Enter the central angle in degrees — the angle measured at the center of the circle between the two radii drawn to the arc's endpoints. The calculator divides that value by 2 to give the inscribed angle that opens onto the same arc from a point on the circumference.
The Formula Explained
The relationship is simple and exact: $$\theta_{\text{inscribed}} = \frac{\text{Central Angle}}{2}$$. Because every inscribed angle that subtends the same arc has the same measure, you only need the central angle to find them all. A direct consequence is Thales' theorem: an angle inscribed in a semicircle (central angle 180°) is always a right angle (90°).
Worked Example
Suppose an arc has a central angle of 120°. The inscribed angle subtending the same arc is $$120 \div 2 = 60°$$ Any point you pick on the major arc will see the chord under this same \(60°\) angle.
FAQ
Does the inscribed angle change if I move the vertex? No — as long as the vertex stays on the same arc and subtends the same chord, the inscribed angle is constant.
What is the inscribed angle for a diameter? A diameter has a central angle of 180°, so the inscribed angle is 90° — a right angle.
Can the central angle exceed 360°? No. A central angle ranges from 0° to 360°, so inscribed angles range from 0° to 180°.
