Inscribed Angle Calculator

Introduction

The inscribed angle theorem is one of the few results in geometry that is both surprising and completely reliable: an angle drawn from a point on a circle to the two ends of a chord is exactly half the central angle that subtends the same arc, no matter where on the circle the point is placed. Slide the vertex along the circumference and the angle does not change. Move the chord and both angles change together, always in the same 2-to-1 ratio [mathworld-inscribed-angle].

That invariance is what makes the theorem useful rather than merely elegant. It means an angle can be measured from anywhere on the circle and still give the same answer, which is exactly the property you want when the centre of a circle is inaccessible — a surveyor standing at the edge of a curved road, an engineer checking a circular flange without being able to reach the middle, or a student given a diagram with the centre marked but not measured. In all of those cases the inscribed angle carries the same information as the central angle, at half the size.

The theorem also produces two consequences that show up constantly in circle problems. First, an angle inscribed in a semicircle is always a right angle, because the arc it subtends is 180° and half of that is 90°. Second, a cyclic quadrilateral — a four-sided figure whose corners all lie on a circle — has opposite angles that sum to 180°, because the two angles subtend complementary arcs. Both results fall straight out of the same half-angle relationship, and both are common in exam questions and in practical layout work.

The direction of the calculation matters as much as the theorem. Sometimes the arc is known and the inscribed angle is wanted; sometimes the angle is measured and the arc is wanted; sometimes the other inscribed angle — the one on the far side of the chord — is wanted, and that is the supplementary value. This calculator handles all three directions from a single input, and shows the arithmetic so the relationship is visible rather than assumed [khan-geo].

It reports the inscribed angle, the central angle, the arc measure, the supplementary inscribed angle, and the check that the two inscribed angles sum to 180°. The Circle Calculator handles the lengths of arcs and chords; this tool handles the angles that the same arcs subtend.

The theorem belongs to a family of circle results that measure the same thing from different vantage points. A vertex on the circle gives the half relationship; a vertex inside the circle gives an angle equal to half the sum of the two arcs it intercepts, which is always larger; and a vertex outside gives half the difference of the arcs, which is always smaller. Seeing the three together makes the inscribed case memorable rather than arbitrary: it is the boundary case in which the two arcs the angle subtends are one and the same arc. That is why the theorem states "vertex on the circumference" as a condition rather than as a detail — move the vertex even slightly and the relationship changes form.

In practice the theorem is used in two directions. When the arc is known — from an angle measured at the centre, or from a fraction of the circumference — the inscribed angle is a halving. When the inscribed angle is known — from a protractor placed anywhere on the rim, which is often the only thing accessible — the arc and the central angle are a doubling. Both directions are the same equation read in opposite order, and both appear in surveying, in drafting, and in the standard set of circle-theorem exercises.

How to Use

Enter either the intercepted arc in degrees or the central angle — they are the same number, since the central angle is defined as the measure of the arc it subtends. The calculator returns the inscribed angle and the related values immediately.

The Formula

For an arc of measure $m$ degrees, the central angle equals the arc and the inscribed angle is half of it:

The supplementary inscribed angle is the one subtending the other arc, the remainder of the circle:

Because the two inscribed angles subtend complementary arcs, they always sum to 180°:

Working backwards, from an inscribed angle $\alpha$ to the arc:

For an arc of 120°: $\text{Inscribed} = 60^\circ$, $\text{Supplementary} = 180 - 60 = 120^\circ$, and the two sum to 180° as required.

Reference Table

The table lists arcs at regular intervals with the angles they subtend from both sides of the chord:

The linearity is the whole theorem in one picture: the inscribed angle is always half the arc, with no curvature, no special cases, and no exceptions. The only structural feature worth marking is the arc of 180°, where the inscribed angle reaches exactly 90° and the two inscribed angles become equal — the semicircle case. Below that the near-side angle is acute and the far-side angle obtuse; above it the roles swap, and the two cross at the right angle [nctm].

Practical Tips

Identify which arc you have. The arc measure and the central angle are the same number, but the inscribed angle is half of it. Most errors in these problems come from applying the halving twice or not at all.

Use the semicircle as a test case. Any time you are told a triangle is inscribed in a circle and one of its sides is a diameter, the angle opposite that side is 90°. This resolves a large fraction of exam problems instantly.

Remember that the two inscribed angles are supplementary. If a problem gives one inscribed angle and asks for the one on the other side of the chord, the answer is simply 180° minus the first. No further geometry is needed.

Check the cyclic quadrilateral rule. If four points lie on a circle, opposite angles sum to 180°. Verifying that in a diagram is a fast way to confirm that the four points really are concyclic.

The theorem needs the vertex on the circle. An angle with its vertex inside the circle but not on the circumference is larger than the inscribed angle; a vertex outside gives a smaller one. Only a vertex exactly on the circle gives the clean half relationship [nist-geo].

Angles are measured along the arc they open onto. When a chord divides a circle into two arcs, each inscribed angle subtends the arc opposite its vertex, not the arc it sits on. Getting that backwards inverts the two supplementary answers.

Use the tool to check a diagram, not just to compute. Entering an arc and comparing the printed inscribed angle against the drawing is often faster than measuring the angle with a protractor, especially for small arcs where protractor error is proportionally large.

Limitations

This calculator works in degrees. Radians are not accepted, because the theorem is most naturally stated in degrees; to convert, multiply by $\pi/180$ and use the Trigonometry Calculator.

It assumes a single circle and a single chord, with the vertex of the inscribed angle exactly on the circumference. Angles with a vertex inside or outside the circle follow different rules (the intersecting chords and intersecting secants theorems) and are not handled here.

The arc measure must be between 0° and 360°. Values outside that range describe a wrap-around that has no distinct geometric meaning, since an arc of 370° is the same as one of 10°.

The calculator reports the supplementary inscribed angle as well, which is the answer to a different question than the primary result. Take care to report the one the problem actually asks for — the acute or the obtuse value.

Finally, the tool deals in angles only. Arc lengths, chord lengths and sector areas are functions of the circle's radius and are handled by the Circle Calculator.

Frequently Asked Questions

❓ What is the inscribed angle theorem?

✅ An angle whose vertex lies on a circle and whose sides pass through the ends of a chord is exactly half the central angle subtending the same arc. For an arc of 120 degrees the inscribed angle is 60 degrees, wherever on the circle the vertex sits.

❓ How do I find the inscribed angle from the arc?

✅ Halve the arc measure. An arc of 80 degrees subtends an inscribed angle of 40 degrees. The central angle equals the arc, so it is 80 degrees.

❓ How do I find the arc from the inscribed angle?

✅ Double the inscribed angle. An inscribed angle of 35 degrees means the arc is 70 degrees and the central angle is also 70 degrees.

❓ Why is an angle in a semicircle always 90 degrees?

✅ The arc of a semicircle is 180 degrees, and the inscribed angle is half of that, so it is 90 degrees. This is Thales's theorem and it holds from any point on the semicircle.

❓ What is the supplementary inscribed angle?

✅ The angle from a vertex on the other side of the chord, subtending the remaining arc. It is 180 degrees minus the first inscribed angle, and the two always add to 180.

❓ Does the position of the vertex change the inscribed angle?

✅ No, as long as the vertex stays on the circle and the chord is unchanged. That invariance is the content of the theorem and the reason a circle's angles can be measured from an accessible point.

❓ What is the relationship between the central angle and the arc?

✅ They are numerically equal. A central angle of 90 degrees intercepts an arc of 90 degrees, which is a quarter of the circle.

❓ What happens if the vertex is inside the circle instead of on it?

✅ The angle is larger than the inscribed angle and equals half the sum of the two arcs it intercepts. Only a vertex exactly on the circumference gives the clean half relationship.

❓ What is the cyclic quadrilateral rule?

✅ If all four corners of a quadrilateral lie on a circle, each pair of opposite angles sums to 180 degrees. It follows from the inscribed angle theorem because the two angles subtend complementary arcs.

❓ Can the inscribed angle be obtuse?

✅ Yes, whenever the arc is greater than 180 degrees. An arc of 240 degrees gives an inscribed angle of 120 degrees, and the supplementary inscribed angle is the acute 60 degrees.

❓ How many degrees is a full circle and how does that help?

✅ A full circle is 360 degrees, so the two arcs formed by any chord sum to 360. That is why the supplementary inscribed angle is (360 - m)/2, which simplifies to 180 - m/2.

❓ Does this calculator work in radians?

✅ No, it works in degrees only. To use radians, convert the angle to degrees first by multiplying by 180/pi, then apply the theorem.