Inscribed Angle Calculator - Theorem-Based Arc to Angle Solver

An inscribed angle calculator applies the inscribed angle theorem to find the angle that sits on the circle when its vertex rests on the circumference and its sides reach two other points on the circle. You give it an intercepted arc, and it returns the inscribed angle in degrees or radians, the central angle that subtends the same arc, and the share of the circle that arc covers.

• • Geometry homework: Solve textbook problems that ask for the inscribed angle when the arc is given, or for the arc when the inscribed angle is given.
• • Circle construction: Lay out a wooden spoke, decorative inlay, or stained-glass segment that opens to a target inscribed angle.
• • Trigonometry preparation: Check the relationship between an inscribed angle and a central angle before moving on to angle-chord or cyclic-quadrilateral work.
• • Navigation and mapping: Convert a bearing or arc distance on a circular chart into the angle that a station on the circumference would measure.

An inscribed angle is the angle whose vertex sits on the circle, with its two sides each passing through another point on the circumference. The arc between those two points is called the intercepted arc, and the inscribed angle is always half of that arc.

The same circle also has a central angle at the center that subtends the same arc. For any arc measured in degrees, the central angle in degrees equals the arc measure exactly, so it is twice the inscribed angle. This double relationship is what the inscribed angle theorem captures, and it is the only rule the calculator needs to produce a numerical answer.

If you already know the central angle and want the chord that the endpoints draw across the circle, the central angle calculator applies the related chord formula and the same radius.

The calculator reads the intercepted arc, divides it by two to get the inscribed angle, and reports the matching central angle alongside it. The central angle for any arc measured in degrees equals the arc measure itself, so it is always exactly twice the inscribed angle.

• interceptedArc: Arc measure in degrees cut off by the inscribed angle. Must be between 0 and 360, exclusive.
• angleUnit: Whether the primary result is shown in degrees or radians. The formula always derives degrees first.

Because the inscribed angle is exactly half of the arc, the formula scales linearly. Doubling the arc doubles the inscribed angle, and halving the arc halves it. The result is exact for any valid arc, which is why inscribed angle problems are a good first example of proportional reasoning in circle geometry.

According to Wolfram MathWorld, an angle inscribed in a circle is exactly half of the central angle that intercepts the same arc, which is the same relationship the calculator uses to derive the inscribed angle from the arc.

The 180-degree arc case then produces the right angle at the semicircle that the calculator reports automatically, with the central angle and the inscribed angle both derived from the same arc measure.

If the intercepted arc is given in a non-degree unit, the angle converter turns it into degrees before you reuse the result with the inscribed angle formula.

These four concepts cover the geometry, the theorem, the arc relationship, and the special semicircle case that the calculator relies on.

The theorem is a direct consequence of the relationship between an inscribed angle and a central angle that share the same arc. Once the central angle is twice the inscribed angle, the arc measured in degrees is also twice the inscribed angle, and the theorem follows.

If the answer needs to switch units outside of this tool, the radians to degrees calculator handles the standalone conversion between the two angle units.

Enter the intercepted arc and pick the answer unit. The calculator returns the inscribed angle plus the supporting circle measurements, with the central angle derived from the arc rather than entered separately.

1. 1 Enter the intercepted arc: Type the arc measure cut off by the inscribed angle. Keep it between 0 and 360 degrees.
2. 2 Pick the answer unit: Choose degrees for protractor work or radians if the angle feeds into a trigonometric formula.
3. 3 Read the inscribed angle: Use the inscribed angle as the primary answer. The central angle, arc fraction, and percent of circle appear as supporting checks.
4. 4 Check the semicircle case: If the arc is 180 degrees, the inscribed angle is exactly 90 degrees regardless of the circle's radius.
5. 5 Verify the central angle: Confirm that the reported central angle equals the arc you entered. The two must match for the inscribed angle to apply.

A stained-glass pattern needs an inscribed angle that opens to 45° on a circle of any size. The intercepted arc is 90° and the central angle is 90° (because the central angle equals the arc in degrees). Enter 90, pick degrees, and the calculator returns an inscribed angle of 45° with the arc covering 25% of the circle.

The arc length calculator extends the same circle relationship when the radius is known and the curve length is the missing quantity.

The main benefit is turning a single arc measurement into the angle that the rest of your geometry needs, without doing the division by hand.

• • Direct arc to angle conversion: Skip the manual division of the arc by two and the unit conversion for the next problem step.
• • Degrees and radians together: See the inscribed angle in both units at once, which is useful when a downstream formula expects radians but the homework key reports degrees.
• • Cross-check the geometry: Central angle, arc fraction, and percent of circle all come from the same arc, so one input set confirms the whole circle.
• • Catch unit mistakes: An arc fraction above 100%, or a percent of circle above 100%, is a fast signal that the inputs were entered in the wrong unit.
• • Drop into other tools: Send the central angle straight into the chord length calculator or the circle calculator without retyping the value.

For homework and exam prep, the inscribed angle calculator removes the most common slip in circle problems: forgetting that the inscribed angle is half the arc. For applied work, the same conversion is what makes the result line up with downstream circle formulas, so the saved time compounds across longer problem sets.

Send the central angle straight into the circle calculator without retyping the value when the same problem also needs the radius, area, or circumference.

The result depends on the quality of the arc input and on whether the answer unit matches the next step in your problem.

• • Assumes a flat circle. Does not handle spherical triangles or great-circle arcs.
• • The intercepted arc must be positive and strictly less than 360 degrees; other inputs are rejected before the formula runs.
• • The central angle shown is derived from the arc, not entered by the user. The two must agree, and the calculator always reports the central angle that matches the arc you entered.

According to Wolfram MathWorld on Thales' Theorem, an inscribed angle in a semicircle is a right angle, which is the most important sanity check for any inscribed angle result near the 180° arc case.

If the same circle problem needs the chord that joins the two endpoints of the arc, the chord length calculator takes the radius and the central angle and returns the straight-line distance across the arc.