Which of the two arcs between the chord endpoints is intercepted?

Use the arc that does not contain the angle's vertex. It can be the minor or major arc, depending on where that vertex lies.

Why does a diameter produce a right inscribed angle?

Its intercepted semicircle measures 180°. Half of that measure is 90°, provided the vertex is another point on the circumference.

Inscribed Angle Calculator

The inscribed angle calculator finds an angle whose vertex lies on a circle and whose sides are chords of that circle. Enter the measure of the intercepted arc to find the angle, or enter the angle to find its arc measure. The highlighted arc is the one between the chord endpoints that does not contain the vertex. Both inputs and outputs on this page use degrees. Physical arc length is a separate distance quantity.

Apply the circle theorem with the inscribed angle calculator

The inscribed-angle theorem states that the angle equals half the measure of its intercepted arc. A central angle covering the same selected arc has the arc's full measure, so the inscribed angle is half that central measure. The vertex position is essential: moving it to the center creates a central angle, and moving it inside or outside the circle can require different chord or secant theorems. A similar-looking V shape alone does not establish the half-arc rule.

Locate the vertex and the intercepted arc first

Check that the vertex and both chord endpoints are three distinct points on the same circle. The drawing marks the vertex V. The two solid chord segments run from V to the endpoints of the highlighted circumference arc.
Choose Intercepted arc to calculate the inscribed angle. Enter a measure strictly between 0° and 360°. An arc labeled 120° is an angular measure; a curve labeled twelve centimeters is a length and belongs in the central-angle workflow instead.
Choose Inscribed angle to calculate the intercepted arc. Enter a value strictly between 0° and 180°. The tool doubles it and redraws the selected arc, including a major arc when the inscribed angle is obtuse.
Read the arc and angle fields together. The central-angle comparison refers to the same highlighted arc, so it can be reflex. The diagram keeps the vertex on the complementary arc and does not substitute a minor arc merely because it looks shorter.

Half the intercepted arc, twice the inscribed angle

If m is intercepted arc measure and α is the inscribed angle, α = m/2. The inverse is m = 2α. These formulas use angular measures directly, so a radius is not needed. Changing a circle's physical size preserves both angles.
The strict domains are 0° < m < 360° and 0° < α < 180°. Endpoints would collapse chord endpoints or the angle into a degenerate configuration. Nonfinite values and values outside these ranges are rejected before a result or diagram is produced.
A semicircle has arc measure 180°, so an inscribed angle intercepting it is 90°. This is the angle-in-a-semicircle result often called Thales' theorem. The chord joining the arc endpoints is a diameter, and the angle's vertex lies elsewhere on the circle.
An inscribed angle of 120° intercepts a major arc of 240°. Its vertex lies on the remaining 120° arc. The central measure for that selected major arc is reflex; the smaller central angle between the same radii would describe the other arc.

Worked examples and the choice of arc

An intercepted arc of 120° gives α = 120°/2 = 60°. The central angle for that arc is 120°. This example distinguishes the two vertex positions while keeping the endpoints and selected arc the same.
An inscribed angle of 47° gives an intercepted arc of 94°. No length measurement is required. If a circle is drawn twice as large, the chords grow longer but the angle and arc measure remain 47° and 94°.
An intercepted arc of 270° produces an inscribed angle of 135°. Selecting the remaining 90° arc instead would produce a 45° angle at a vertex on the other side. Always identify which arc excludes the vertex before doing the division.
Two vertices placed on the same complementary arc and joined to the same endpoints intercept the same arc. Their inscribed angles are equal even when their chord lengths differ. Vertices on opposite sides intercept complementary arcs, so the resulting angles add to 180° in a cyclic quadrilateral.

Classroom uses and the boundary of the theorem

This tool helps learners check circle-theorem exercises, compare minor and major arcs, and understand why an angle's location matters more than the drawing's size. Teachers can switch between forward and inverse modes to ask for the missing arc rather than always asking for the angle. A tangent and chord can have another useful half-arc relationship, but that is a different configuration from the two chords drawn here. Intersecting chords inside a circle and secants outside it likewise require additional arc information and are not solved by this page.

Arc measure is not a curved distance

Do not enter a circumference distance into a degree field. To convert length into arc measure, first use the central angle calculator with a known radius, then apply the half-arc theorem. The inscribed angle calculator works with browser floating-point values and displays up to six decimal places for ordinary numbers; extreme values use six significant digits and endpoint values may retain extra digits. Decimal outputs are labeled approximate in the tool. Very small arcs can make the chord endpoints hard to distinguish on screen, but the textual values remain available. The drawing is an illustration of the stated geometry, not a physical measuring scale.