How this instrument works
A central angle sits at a circle's own center, its two arms running out along radii to the ends of an arc. An inscribed one looks at that same arc from a point on the circle's rim instead. The Inscribed Angle Theorem ties the two together with a fixed ratio: inscribed = central ⁄ 2, no matter which arc or which circle. The proof sketch is short — draw the radius from the center out to the inscribed vertex, and it splits the figure into two isosceles triangles built from pairs of equal radii; the exterior angle theorem shows each triangle's share of the center is exactly twice its own base, and the two bases together equal the inscribed value itself.
The genuinely surprising part is what the theorem does not depend on. Slide the rim vertex anywhere else along the remaining arc, and the reading it produces toward the same two endpoints stays fixed — every point on that longer arc sees the shorter one at an identical span. A lighthouse keeper, a sailor, and a surveyor standing at three different spots on a harbor's curved seawall, all sighting the same stretch of far shoreline, would each measure the same figure between their sightlines, purely because they share one circle.
The relationship has a famous limiting case built in. Push the central reading to a straight 180°, so the two radii form a diameter, and the inscribed value lands on exactly 90° — this is Thales' theorem, and it is why any triangle drawn from the two ends of a diameter to a third point anywhere on the circle is automatically a right triangle. Push the central figure to the full 360° and the inscribed one reaches 180°, a degenerate case: the 'triangle' has flattened into a straight line, since the vertex, viewed across a whole circle, sees no arc left to define one.
- Enter the figure measured at the circle's center into Central angle, choosing deg, rad, or turn from the unit control beside the field.
- Read Inscribed angle for the result — it reports exactly half of whatever Central angle holds, converted into whichever unit you're viewing it in.
- Try the default of 80° first: Inscribed angle should land on exactly 40° for the same arc.
- Switch either field's unit to rad or turn to confirm the halving is unit-independent — 180°, π rad, and 0.5 turn all fold to the same 90° ⁄ π⁄2 rad ⁄ 0.25 turn inscribed reading.
Worked example — an 80° central angle
Picture two spokes from the center of a circular plaza fixed 80° apart, marking off one arc of the rim. Set Central angle to 80° and the sheet returns Inscribed angle = central ⁄ 2 = 80 ⁄ 2 = 40°, or, carried in the radians the engine actually computes with, 1.3962634015954636 ⁄ 2 = 0.6981317007977318 — 40° to eight decimal places. Anyone standing anywhere else on the plaza's rim, looking across at that same 80° arc, measures an identical 40° span between their two sightlines to its ends.
The theorem does not care where the observer stands, only that they stand on the longer arc looking across at the shorter one. Move a viewer a few meters along the rim and their sightlines to the same two rim points swing, but the span between those sightlines holds at 40° until the vertex itself crosses onto the arc being measured. That invariance is exactly what makes the ratio central ⁄ 2 trustworthy as one fixed number rather than a range that depends on where someone happens to stand.
Questions
What does the Inscribed Angle Theorem actually say?
Any angle formed at a point on a circle's rim, looking across at an arc, equals exactly half of whatever that same arc makes at the circle's center. Move the rim point anywhere along the remaining arc and the reading stays fixed — only the arc being viewed decides its size, never the viewer's exact position.
Why is the inscribed angle always exactly half, never some other fraction?
Draw a radius from the center out to the inscribed vertex and it splits the figure into two isosceles triangles built from pairs of equal radii. The exterior angle theorem shows each triangle contributes a central share exactly twice its own base; add the two bases together and the half falls out directly, not as an approximation.
What is Thales' theorem, and how does it relate to this calculator?
Thales' theorem is the special case where the central angle is a straight 180°, meaning the two radii form a diameter. Half of 180° is 90°, so a triangle drawn from the two ends of a diameter to any third point on the circle is automatically a right triangle, whichever point is chosen.
What is the most common mistake people make with this relationship?
Halving the wrong reading, or forgetting that both figures must sit on the same arc. It is easy to instead measure a chord's angle against a tangent line at the point of contact — a related but different relationship — or to read off the reflex central figure on the far side of the circle by mistake, which doubles the correct answer instead of halving it.
Does the formula still work past a central angle of 180°?
Yes, provided the reading describes the actual arc swept. Halving a 300° central angle correctly gives a 150° inscribed value for the corresponding major arc. The relationship only stops describing a real circle once the central figure passes 360°, a full revolution.
How is this different from the central angle calculator on this site?
The central angle sheet solves a separate question entirely — arc length divided by radius, θ = s ⁄ r — with no second angle involved anywhere. This sheet starts from a central reading already known, in any unit, and reports the inscribed value for that same arc, or runs backward from an inscribed figure by doubling it.