Inscribed Angle Calculator — Inscribed = ½ × Central Angle
Apply the Inscribed Angle Theorem to find the inscribed angle from the central angle (or arc), or the central angle from the inscribed angle. Choose what you know and enter one value — the other is calculated instantly.
Solve for
°
Inscribed angle = ½ × central angle
- 1
Given central angle
80 - 2
Inscribed angle = central ÷ 2
80 ÷ 2 = 40The inscribed angle is always half the central angle subtending the same arc.
How does this calculator work?
The inscribed angle is always half the central angle subtending the same arc: inscribed = central / 2. Special case: an angle inscribed in a semicircle (arc = 180°) is 90° — Thales' theorem. Enter either value to find the other instantly.
Formula
How this is calculated
The Inscribed Angle Theorem states that an angle formed at the circumference of a circle (inscribed angle) is always exactly half the central angle that subtends the same arc. The central angle has its vertex at the centre of the circle; the inscribed angle has its vertex anywhere on the major arc — both subtending the same chord.
Equivalently, because the intercepted arc in degrees equals the central angle, the inscribed angle is also half the intercepted arc. This relationship holds regardless of where on the major arc the inscribed angle's vertex sits — all inscribed angles that subtend the same arc are equal. A special and famous case is Thales' theorem: an inscribed angle subtending a diameter (arc = 180°) is always a right angle (90°).
The theorem is foundational to circle geometry and appears in proofs about cyclic quadrilaterals, angle bisectors, and tangent-chord angles. It also underlies applications in optics (angle of view for a lens), navigation (equal-angle loci for cross-bearing fixes), and architectural geometry.
Frequently asked questions
Both angles subtend the same arc. A geometric proof places the inscribed angle vertex on the circle, draws a diameter from it, then uses the isoceles triangles formed by two radii to show that the central angle equals the sum of two base angles — exactly twice the inscribed angle.
Thales' theorem is the special case where the inscribed angle subtends a diameter (a 180° arc). The central angle is 180°, so the inscribed angle is 90° — a right angle. It follows that any triangle inscribed in a semicircle with the diameter as hypotenuse is a right triangle.
Yes. All inscribed angles with their vertex on the major arc that subtend the same chord are equal, regardless of exactly where the vertex sits on that arc. This corollary is often used in proofs about cyclic quadrilaterals and tangent-chord angles.
Also known as
TG we-Calculate Editorial Team. (2026). Inscribed Angle Calculator — Inscribed = ½ × Central Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/inscribed-angle-calculator
TG we-Calculate Editorial Team. "Inscribed Angle Calculator — Inscribed = ½ × Central Angle." TG we-Calculate. 2026. https://we-calculate.com/calculator/inscribed-angle-calculator.
TG we-Calculate Editorial Team, "Inscribed Angle Calculator — Inscribed = ½ × Central Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/inscribed-angle-calculator
@misc{wecalculate_inscribed_angle_calculator, title = {Inscribed Angle Calculator — Inscribed = ½ × Central Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/inscribed-angle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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