Calculate inscribed angles, central angles, arc length, chord length, sagitta, sector area, and segment area for a circle. Includes Thales' theorem detection and a reference table.
An inscribed angle is an angle formed by two chords that share an endpoint on a circle. The Inscribed Angle Theorem states that an inscribed angle is always exactly half the central angle that subtends the same arc. This elegant relationship is one of the cornerstones of circle geometry.
Formally, if a central angle measures θ degrees, the inscribed angle on the same arc is θ/2 degrees. Conversely, if you know the inscribed angle α, the central angle is 2α. The intercepted arc has a degree measure equal to the central angle.
A famous special case is Thales' theorem: when the inscribed angle subtends a diameter (central angle = 180°), the inscribed angle is exactly 90°. This means any triangle inscribed in a semicircle with the diameter as one side is a right triangle.
Beyond angles, the arc, chord, and sagitta are all related. The arc length is rθ (in radians). The chord length is 2r sin(θ/2). The sagitta (the height of the circular segment) is r(1 − cos(θ/2)). Sector area is ½r²θ and segment area adds the triangle subtraction.
This calculator accepts a central angle, inscribed angle, or arc length with radius, and computes all related quantities. It flags Thales' theorem when it applies, provides a reference table of common angle pairs, and visualizes the proportional relationships with bars.
The Inscribed Angle Calculator — Central Angle, Arc & Chord is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Central Angle, Inscribed Angle, Arc Length in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Inscribed Angle = ½ × Central Angle Central Angle = 2 × Inscribed Angle Arc Length = r × θ (θ in radians) Chord Length = 2r × sin(θ/2) Sagitta = r × (1 − cos(θ/2)) Sector Area = ½r²θ Segment Area = ½r²(θ − sin θ) Thales' Theorem: inscribed angle = 90° when subtending a diameter
Result: Inscribed Angle = 45°, Arc ≈ 15.708, Chord ≈ 14.142, Sagitta ≈ 2.929
For central angle 90° and r = 10: Inscribed = 90/2 = 45°. Arc = 10 × π/2 ≈ 15.708. Chord = 2 × 10 × sin(45°) = 20 × 0.7071 ≈ 14.142. Sagitta = 10 × (1 − cos 45°) = 10 × 0.2929 ≈ 2.929.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Inscribed Angle Calculator — Central Angle, Arc & Chord calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
It states that an inscribed angle is always half the central angle that intercepts the same arc. If the central angle is θ, the inscribed angle is θ/2.
A special case of the inscribed angle theorem: any angle inscribed in a semicircle (subtending a diameter) is exactly 90°. This forms a right triangle.
Yes. If the central angle exceeds 180°, the inscribed angle exceeds 90°. For example, a 240° central angle gives a 120° inscribed angle.
First double the inscribed angle to get the central angle θ, then compute chord = 2r × sin(θ/2). Use this as a practical reminder before finalizing the result.
The sagitta (or versine) is the distance from the midpoint of a chord to the midpoint of its arc. It equals r(1 − cos(θ/2)).
Yes. The inscribed angle theorem holds for all circles, regardless of size. The radius affects lengths (arc, chord, sagitta) but not the angle relationship.