The inscribed angle calculator above applies the inscribed angle theorem: an angle with its vertex on a circle is exactly half the central angle standing on the same arc. Give it any one of the inscribed angle, the central angle, the arc measure or the tangent-chord angle and it returns the rest, and with a radius it adds the arc length and the chord length.
Arb Digital publishes it as the angle-theorem page. The boundary against our other circle tools is worth stating: the circle calculator converts between radius, diameter, circumference and area; the sector area calculator and the arc length calculator give the measurements of a slice once you already know its angle. None of them relates one angle to another. This page does only that — it works out an angle you do not know from an angle you do, and the measurements come along afterwards.
What This Inscribed Angle Calculator Does
It implements four relationships, all consequences of the same theorem. The central angle is twice the inscribed angle standing on the same arc. The intercepted arc measure in degrees equals the central angle, by definition of arc measure. The tangent-chord angle, formed between a chord and the tangent at one of its endpoints, is also half the intercepted arc, which makes it equal to any inscribed angle on that arc. And the opposite angle in a cyclic quadrilateral is 180 degrees minus the inscribed angle, which the tool reports alongside.
Where a radius is supplied it converts the arc measure to radians and multiplies by the radius for the arc length, and computes the chord as twice the radius times the sine of half the central angle. Both are reported in whatever unit the radius was given in.
Degenerate and boundary cases get a written explanation rather than a silent number. An inscribed angle of exactly 90 degrees means the central angle is 180 and the chord is a diameter, which is Thales's theorem. An inscribed angle at or above 180 is impossible and is refused. So is a central angle above 360.
How to Use It
- Choose which quantity you know. The calculator solves in every direction, so it does not matter which one you have.
- Enter it in degrees. Radians are not accepted as input, but the working line shows the radian conversion used for the arc length.
- Add the radius if you want lengths. Without it the angles are still fully determined, since the theorem is scale-free.
- Read the extra relationships in the panel below the grid — the cyclic quadrilateral angle and the major-arc figures are frequently what a problem actually asks for.
- Watch the boundary cases. An inscribed angle of exactly 90 degrees is a semicircle and is flagged as such.
The Formula and How It Is Calculated
The inscribed angle theorem states that central angle = 2 × inscribed angle, or equivalently that the inscribed angle is half the intercepted arc. Arc length is L = r × θ with θ in radians, and the chord is c = 2r sin(θ/2) where θ is the central angle.
Work the default through by hand. An inscribed angle of 35 degrees gives a central angle of 2 × 35 = 70 degrees, and the intercepted arc therefore measures 70 degrees as well. Converting, 70 degrees is 70π ÷ 180 = 1.221730 radians, so with a radius of 10 the arc length is 10 × 1.221730 = 12.21730 units.
The chord is 2 × 10 × sin(35°) = 20 × 0.573576 = 11.47153 units. Note that the half-angle in the chord formula is half of the central angle, which is the inscribed angle itself — a small coincidence of this construction that makes the chord formula easy to remember here. The opposite angle in a cyclic quadrilateral through the same chord is 180 − 35 = 145 degrees, and the major arc measures 360 − 70 = 290 degrees. Wolfram MathWorld's page on the inscribed angle states the doubling relationship, and its page on the central angle gives the same result from the other side.
Why the Vertex Can Move Without Changing the Angle
The most useful consequence of the theorem is one that surprises people the first time: every inscribed angle standing on the same arc is equal. Slide the vertex anywhere along the major arc and the angle it subtends over the chord does not change at all.
The reason follows immediately. Each of those angles is half the same central angle, and the central angle is fixed once the two endpoints are fixed. So they must all be equal to each other. This is often called the angles in the same segment theorem, and it is the engine behind a great many geometry proofs — whenever you need two angles to be equal and both sit on a circle over the same chord, you are done without further work.
Thales's theorem is the special case worth memorising. When the chord is a diameter the central angle is a straight 180 degrees, so every inscribed angle on it is 90. That means any triangle with one side a diameter and its third vertex anywhere on the circle is right-angled, and conversely the hypotenuse of any right triangle is a diameter of its circumcircle. Press the Thales preset to see the tool report exactly that.
Which Arc, and Why It Matters
Two points on a circle define two arcs, a minor one and a major one, and almost every mistake with this theorem is a mistake about which of them is intended.
The rule is that the intercepted arc is the one lying inside the inscribed angle, on the far side of the chord from the vertex. If your vertex sits on the major arc, the intercepted arc is the minor one, and the inscribed angle is acute or right. Move the vertex to the minor arc and the intercepted arc becomes the major one, so the angle is obtuse. That is why the two possible inscribed angles over a given chord — one from each side — always add to 180 degrees, which is exactly the cyclic quadrilateral property.
The tool reports both arcs so this stays visible. If your answer is out by exactly 180 degrees, or you have an acute angle where a problem clearly shows an obtuse one, the arc is where to look. Our chord calculator works between chord length, radius and the central angle if you are coming at the same figure from the measurement side, and the unit circle calculator is the reference for the trigonometric values involved.
The Related Angle Theorems It Sits Beside
The inscribed angle theorem is the middle case of a family, and knowing the neighbours prevents applying the wrong one.
When the vertex is at the centre, the angle equals the arc. When it is on the circle, the angle is half the arc — this theorem. When it is inside the circle but not at the centre, formed by two intersecting chords, the angle is half the sum of the two arcs the vertical pair intercepts. When it is outside the circle, formed by two secants, two tangents, or one of each, the angle is half the difference of the two intercepted arcs.
The pattern is worth seeing whole: centre gives the arc, on gives half, inside gives half the sum, outside gives half the difference. The tangent-chord angle fits in as the limiting case of the inscribed angle where one side of the angle rotates until it touches the circle at a single point, which is why it takes the same value. For work with the tangent line itself rather than the angle, our tangent line to circle calculator handles the coordinate geometry.
Arb Digital builds free tools that show the theorem behind the answer and flag the boundary cases.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Doubling in the wrong direction — the central angle is twice the inscribed one, not the other way round, and reversing it is the single most common slip.
- Taking the wrong arc — the intercepted arc is the one inside the angle, away from the vertex, and choosing the other gives an answer 180 degrees out.
- Using the theorem when the vertex is not on the circle — inside gives half the sum of two arcs and outside gives half the difference, which are different rules.
- Halving the wrong angle for the chord — the chord formula uses half the central angle, which happens to equal the inscribed angle, so it is easy to halve twice by accident.
- Mixing degrees and radians — arc length needs radians, and using degrees directly gives a length wrong by a factor of about 57.
Related Free Tools From Arb Digital
Convert between radius, diameter, circumference and area with the circle calculator, measure a slice with the sector area calculator or the arc length calculator, work between chord and radius with the chord calculator, find a tangent in coordinates with the tangent line to circle calculator, or look up exact values with the unit circle calculator. The full free online tools hub lists every geometry tool we publish.
Frequently Asked Questions
An angle whose vertex lies on a circle is exactly half the central angle standing on the same arc. Equivalently, the inscribed angle is half the degree measure of the arc it intercepts.
Because each is half the same central angle, and that central angle is fixed once the two endpoints of the arc are fixed. Moving the vertex along the circle changes nothing.
The special case where the chord is a diameter. The central angle is then 180 degrees, so every inscribed angle on it is 90, meaning any triangle on a diameter with its third vertex on the circle is right-angled.
It is the arc lying inside the angle, on the opposite side of the chord from the vertex. Taking the other arc gives an answer that is out by 180 degrees.
The angle between a chord and the tangent drawn at one of its endpoints. It is half the intercepted arc, which makes it equal to any inscribed angle standing on that same arc.
Yes, whenever the vertex sits on the minor arc so that the intercepted arc is the major one. The two inscribed angles over a chord, one from each side, always add to 180 degrees.
A vertex inside the circle gives half the sum of the two intercepted arcs; a vertex outside gives half the difference. Only a vertex on the circle gives the simple halving.
Not for the angles, because the theorem is scale-free. The radius is needed only for the arc length and the chord length, which are actual measurements.
This page explains a result of plane Euclidean geometry for educational purposes. Angles are computed in degrees and converted internally to radians, so displayed values are rounded to the number of decimal places you choose.
