Inscribed Angle Calculator

Enter either the measure of the intercepted arc or the central angle to find the inscribed angle.

Intercepted Arc Measure (degrees):

Central Angle Measure (degrees):

Understanding the Inscribed Angle

An inscribed angle is a fundamental concept in geometry, particularly when dealing with circles. It is defined as an angle formed by two chords in a circle that have a common endpoint on the circle's circumference. This common endpoint is the vertex of the inscribed angle.

The Inscribed Angle Theorem

The most crucial property of an inscribed angle is described by the Inscribed Angle Theorem. This theorem states that the measure of an inscribed angle is exactly half the measure of its intercepted arc. The intercepted arc is the portion of the circle's circumference that lies between the two endpoints of the chords forming the angle, not including the vertex.

For example, if an inscribed angle intercepts an arc of 120 degrees, the measure of the inscribed angle will be 60 degrees (120 / 2).

Relationship to Central Angles

Another way to understand inscribed angles is through their relationship with central angles. A central angle is an angle whose vertex is the center of the circle and whose sides are radii intersecting the circle at two points. If an inscribed angle and a central angle subtend (intercept) the same arc, then the measure of the inscribed angle is half the measure of the central angle.

For instance, if a central angle measures 100 degrees, and an inscribed angle intercepts the same 100-degree arc, the inscribed angle will be 50 degrees (100 / 2).

Key Corollaries and Properties:

Angles Subtending the Same Arc: All inscribed angles that subtend the same arc are equal in measure.
Angle in a Semicircle: An inscribed angle that intercepts a semicircle (an arc of 180 degrees) is always a right angle (90 degrees). This is because 180 / 2 = 90.
Cyclic Quadrilaterals: In a cyclic quadrilateral (a quadrilateral whose vertices all lie on a single circle), opposite angles are supplementary (add up to 180 degrees).

How to Use the Inscribed Angle Calculator

Our Inscribed Angle Calculator simplifies the process of finding the measure of an inscribed angle. You only need one piece of information:

Intercepted Arc Measure: If you know the measure of the arc that the inscribed angle "cuts off" from the circle, enter it into the "Intercepted Arc Measure (degrees)" field.
Central Angle Measure: Alternatively, if you know the measure of the central angle that subtends the same arc as the inscribed angle, enter it into the "Central Angle Measure (degrees)" field.

The calculator will automatically apply the Inscribed Angle Theorem to provide you with the correct angle measure in degrees. If you enter both values, the calculator will prioritize the intercepted arc measure for the calculation.

Examples:

Example 1: Using Intercepted Arc

Suppose an inscribed angle intercepts an arc that measures 110 degrees. To find the inscribed angle:

Enter "110" into the "Intercepted Arc Measure (degrees)" field.
Click "Calculate Inscribed Angle".
The result will be 55.00 degrees (110 / 2).

Example 2: Using Central Angle

Consider a central angle that measures 75 degrees, and an inscribed angle subtends the same arc. To find the inscribed angle:

Enter "75" into the "Central Angle Measure (degrees)" field.
Click "Calculate Inscribed Angle".
The result will be 37.50 degrees (75 / 2).

This calculator is a handy tool for students, educators, and anyone working with circle geometry, making complex calculations straightforward and accurate.