Inscribed Angle Calculator

Enter a circle's central angle (and optionally its radius) to find the inscribed angle using the Inscribed Angle Theorem, plus the arc length.

Quick Facts

Inscribed Angle Theorem
θ_inscribed = θ_central / 2
An inscribed angle is always half the central angle that subtends the same arc.
Thales' Theorem
90° for a diameter
An inscribed angle that subtends a semicircle (180° central angle) is always a right angle.
Cyclic Quadrilateral Rule
Opposite angles sum to 180°
Inscribed angles on opposite arcs of the same chord are supplementary.

Your Results

Calculated
Inscribed Angle (this arc)
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θ_inscribed = central angle ÷ 2
Inscribed Angle (remaining arc)
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Supplement: 180° − (central angle ÷ 2)
Arc Length
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s = r × θ (θ in radians)
Angle Classification
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Acute, right, or obtuse

Ready

Enter a central angle (and radius, if you want the arc length), then press Calculate.

Formula and Method for the Inscribed Angle Calculator

An inscribed angle is formed by two chords of a circle that share an endpoint — the vertex — on the circle itself. The Inscribed Angle Theorem states that an inscribed angle is always exactly half of the central angle that subtends the same arc: θ_inscribed = θ_central / 2. This calculator takes the central angle of an arc (and, optionally, the circle's radius) and returns the inscribed angles on both sides of that chord, along with the arc length.

How the calculation works

Enter the central angle θ formed at the circle's center by the two radii running to the endpoints of an arc (0° < θ < 360°). The inscribed angle measured from a point on the opposite arc is θ/2. Because the vertex and the two chord endpoints, together with any point on the other arc, form a cyclic quadrilateral, the inscribed angle measured from a point on the near side — looking across at the remaining arc of 360° − θ — is its supplement: 180° − θ/2. The two results always add up to 180°. If you also enter a radius, the calculator finds the arc length using s = r × θ_rad, where θ_rad = θ × π/180 converts the central angle from degrees to radians.

Common mistakes

  • Confusing the inscribed angle with the central angle: the central angle is measured at the circle's center; the inscribed angle is measured at a point on the circle's circumference and is always half as large for the same arc.
  • Forgetting which arc the angle subtends: an inscribed angle only equals half the central angle of the arc it "opens onto" — a point sitting on the other side of the chord sees the supplementary angle instead.
  • Mixing degrees and radians: convert to radians (θ × π/180) before using θ in the arc-length formula s = rθ; plugging a degree value directly into s = rθ gives a wildly wrong length.

Real-world applications and special cases

  • Thales' theorem is the special case where the central angle is exactly 180° (the chord is a diameter): any inscribed angle subtending that diameter is exactly 90°. This is used to construct right angles and to test whether a triangle is right-angled by checking if its hypotenuse lies on a circle's diameter.
  • Surveying and navigation use inscribed-angle relationships to fix a position from bearings taken to two known landmarks.
  • Architecture and structural design use the theorem to lay out circular arches, domes, and rounded elements with predictable, repeatable angles.
  • Billiards, optics, and other reflection problems use the same circle geometry to predict angles off a curved surface.

Frequently Asked Questions

What is the Inscribed Angle Theorem?
The Inscribed Angle Theorem states that an inscribed angle is always half of the central angle that subtends the same arc: θ_inscribed = θ_central / 2. For example, a 100° central angle produces a 50° inscribed angle from a point on the opposite arc.
What is Thales' theorem and how does it relate to the inscribed angle?
Thales' theorem is the special case of the Inscribed Angle Theorem where the arc is a semicircle. If the central angle is 180° (the chord is a diameter), any inscribed angle subtending that diameter is exactly 90°.
Do all inscribed angles subtending the same arc equal each other?
Yes. Every inscribed angle whose vertex lies on the same arc and subtends the same chord has an identical measure, because each one equals half of the same central angle.
How do I find the arc length from the central angle and radius?
Convert the central angle to radians (θ × π/180) and multiply by the radius: arc length s = r × θ_rad. This calculator performs that conversion automatically when you supply a radius.

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