Circle Theorems Calculator — Inscribed Angles, Arcs & Chords
Enter a radius and central angle to apply the key circle theorems: find the inscribed angle, arc length, chord, sector and segment areas, tangent-chord angle, and the opposite angle in a cyclic quadrilateral.
units
°
An angle on the circle subtending the same arc = half the central angle
- 1
Central angle to radians
80° × π ÷ 180 = 1.396263 - 2
Inscribed angle
80 ÷ 2 = 40Inscribed Angle Theorem: an angle on the circle subtending the same arc is exactly half the central angle.
How does this calculator work?
Inscribed angle = central angle ÷ 2 (Inscribed Angle Theorem). Arc = rθ_rad, chord = 2r·sin(θ/2), sector area = r²θ/2, segment = sector − triangle. Tangent-chord angle = inscribed angle. Opposite cyclic-quad angle = 180° − inscribed angle.
Formula
How this is calculated
The Inscribed Angle Theorem is the central result of circle geometry: an angle formed by two chords meeting on the circle (an inscribed angle) is exactly half the central angle that subtends the same arc. So if the central angle is 80°, any inscribed angle on the major arc is 40°. This is why every angle in a semicircle is 90° — the central angle for a diameter is 180°, half of which is 90°.
From the same central angle θ (in radians = θ° × π/180) the calculator derives the arc length (r × θ), chord length (2r × sin(θ/2)), sector area (r²θ/2), and segment area (sector minus the isoceles triangle formed by the two radii and chord). The tangent-chord angle — the angle between a tangent drawn at one arc endpoint and the chord — equals the inscribed angle in the alternate segment (the Tangent-Chord Theorem).
For cyclic quadrilaterals (four vertices on a circle), opposite interior angles are supplementary: they sum to 180°. The calculator shows the "opposite" angle as 180° minus the inscribed angle.
Frequently asked questions
An inscribed angle (formed by two chords meeting on the circle) is always half the central angle subtending the same arc. If the central angle is 100°, all inscribed angles on the opposite arc are exactly 50°, regardless of where on that arc the vertex sits.
The diameter subtends a central angle of 180°. By the Inscribed Angle Theorem, any angle inscribed in the semicircle (with the diameter as its subtending chord) is 180°/2 = 90°. This is Thales' Theorem, one of the earliest proved theorems in Greek mathematics.
A cyclic quadrilateral has all four vertices lying on a single circle. Its key property: each pair of opposite interior angles sums to 180°. This calculator shows the opposite angle for an inscribed angle derived from the central angle you enter.
Also known as
TG we-Calculate Editorial Team. (2026). Circle Theorems Calculator — Inscribed Angles, Arcs & Chords [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/circle-theorems-calculator
TG we-Calculate Editorial Team. "Circle Theorems Calculator — Inscribed Angles, Arcs & Chords." TG we-Calculate. 2026. https://we-calculate.com/calculator/circle-theorems-calculator.
TG we-Calculate Editorial Team, "Circle Theorems Calculator — Inscribed Angles, Arcs & Chords," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/circle-theorems-calculator
@misc{wecalculate_circle_theorems_calculator, title = {Circle Theorems Calculator — Inscribed Angles, Arcs & Chords}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/circle-theorems-calculator}}, year = {2026}, note = {TG we-Calculate} }
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