Inscribed Angle Calculator – Theorem, Formula, and Examples
An inscribed angleis an angle formed by two chords of a circle that meet at a single point on the circle's circumference. That meeting point is called the vertex, and the angle "opens up" toward an arc on the far side of the circle called the intercepted arc. Inscribed angles show up throughout geometry, trigonometry, engineering, and design — anywhere circular shapes and angles intersect.
The Inscribed Angle Theorem
The relationship between an inscribed angle and its intercepted arc is governed by one of the most useful results in circle geometry, the Inscribed Angle Theorem:
Inscribed Angle = Intercepted Arc / 2
Equivalently, since the central angle subtending an arc always equals the arc's degree measure, the theorem can also be stated as:
Inscribed Angle = Central Angle / 2
This holds no matter where the vertex sits on the major arc — every inscribed angle that intercepts the same arc has the exact same measure. Rearranging the formula also lets you go the other direction: Intercepted Arc = 2 × Inscribed Angle.
Four Calculation Modes
This calculator supports four common scenarios, selectable from the mode dropdown:
- Arc → Inscribed Angle — enter the intercepted arc to find the inscribed angle. Example: Arc = 100° gives Inscribed Angle = 50°.
- Inscribed Angle → Arc — enter the inscribed angle to find the intercepted arc. Example: Inscribed Angle = 35° gives Arc = 70°.
- Central Angle → Inscribed Angle — enter the central angle to find the inscribed angle subtending the same arc. Example: Central Angle = 120° gives Inscribed Angle = 60°.
- Inscribed Angle → Central Angle — enter the inscribed angle to find the corresponding central angle. Example: Inscribed Angle = 45° gives Central Angle = 90°.
The Semicircle Special Case (Thales' Theorem)
A particularly elegant result falls directly out of the theorem: if the intercepted arc is exactly 180° (a semicircle), then the inscribed angle is always 90°, regardless of where the vertex is placed on the remaining half of the circle. This special case is known as Thales' Theorem and is a frequent shortcut in geometric proofs, right-triangle constructions, and CAD work. The calculator automatically detects this case and displays an explanatory note whenever the arc equals 180°.
Optional Arc Length
If you know the circle's radius, the calculator can also compute the length of the intercepted arc using the standard arc length formula:
Arc Length = (Arc in degrees / 360) × 2πr
Simply enter a radius and choose your preferred unit (mm, cm, m, km, in, or ft) — the arc length field is optional and can be left blank if you only need the angle relationships.
Degrees or Radians
All angle results — inscribed angle, intercepted arc, and central angle — can be toggled between degrees and radians using the output unit selector. Internally the calculator always converts using radians = degrees × (π / 180) to keep results consistent regardless of which unit you view.
Real-World Applications
Inscribed angles are not just a classroom exercise — they appear in many practical settings:
- Wheel and gear design — spoke and tooth placement often relies on inscribed and central angle relationships.
- Architecture — arches and circular windows use inscribed angle properties to determine structural curves.
- Optics — lens and mirror geometry frequently involves angles subtended by arcs of curved surfaces.
- Navigation and surveying — bearing and sighting problems on circular paths use the same arc–angle relationships.
- Geometry proofs — cyclic quadrilaterals, tangent-chord angles, and circle theorems all build on the Inscribed Angle Theorem.
Why Use This Calculator
Rather than manually rearranging the formula each time, this tool lets you pick exactly what you know and instantly see every related value — inscribed angle, intercepted arc, central angle, and optional arc length — along with a live diagram and step-by-step derivation so you can verify the math at a glance.