A circle creates two main kinds of angles: a central angle — with its vertex at the center, and an inscribed angle — with its vertex on the circle itself. The relationship between them is simple: a central angle equals the arc it subtends, and an inscribed angle equals half of that arc. This page covers definitions, the inscribed angle theorem with all its corollaries, clear diagrams, step-by-step worked examples, and an interactive trainer where you can test yourself right away.
Types of angles in a circle: central and inscribed
Draw two segments from a single point inside a circle — you get an angle. Depending on where its vertex lies, there are two main types of angles in a circle:
- Central angle — an angle with its vertex at the center \(O\) of the circle. Its sides are radii.
- Inscribed angle — an angle whose vertex lies on the circle, with sides that cross it (that is, they are chords).
Each of these angles subtends an arc — the part of the circle that lies inside the angle. It is the arc that connects the central and inscribed angles: both are expressed through its degree measure.
If the vertex of an angle ends up inside the circle (but not at the center) or outside it, it is neither an inscribed nor a central angle — such angles have their own formulas, covered at the end of the page.
∠AOC = arc AC = 120°
∠ABC = ½ arc AC = 60°
The inscribed angle theorem and the measure of an inscribed angle
A central angle equals the arc it subtends — this is how the degree measure of an arc is defined. If \(\angle AOC = 120°\), then arc \(AC\) also equals \(120°\).
The inscribed angle theorem: an inscribed angle is measured by half the arc it subtends:
From here follows the key relationship between inscribed and central angles: an inscribed angle is half the central angle that subtends the same arc:
The measure of an inscribed angle does not depend on exactly where its vertex sits on the circle — only the arc it subtends matters. That's why the diagram below is worth remembering as the "master picture" of the whole topic: a green arc with two angles looking at it.
Inscribed and central angles: properties and corollaries
The inscribed angle theorem directly leads to properties that most textbook problems rely on:
- Inscribed angles subtending the same arc are equal. Each of them equals half of the same arc — so they are equal to each other, no matter where their vertices lie.
- An inscribed angle subtending a diameter is a right angle. A diameter splits the circle into two semicircles of \(180°\) each, so such an angle equals \(180° : 2 = 90°\). The converse is also true: if an inscribed angle is a right angle, it subtends a diameter.
- An inscribed angle can be obtuse. If the arc is greater than \(180°\), the angle is greater than \(90°\) — the "half the arc" formula works without exceptions.
These properties also work "in reverse": for example, equal angles subtending the same segment are used to prove that four points lie on the same circle.
Inscribed and circumscribed angles: a quadrilateral in a circle
If all four vertices of a quadrilateral lie on a circle, it is called cyclic (and the circle is circumscribed about it). The angles of such a quadrilateral are inscribed angles, so the same theorem applies to them.
- The sum of opposite angles of a cyclic quadrilateral is \(180°\). Angles \(A\) and \(C\) subtend two arcs that together make up the whole circle (\(360°\)), so \(\angle A + \angle C = 360° : 2 = 180°\).
- An exterior angle of a cyclic quadrilateral equals the interior angle at the opposite vertex. An exterior angle is supplementary to its own interior angle, exactly like the opposite angle is.
- The sum-of-angles property also works as a test: if a quadrilateral has both pairs of opposite angles summing to \(180°\), a circle can be circumscribed about it. That's why any rectangle can be inscribed in a circle, while a rhombus (other than a square) cannot.
Alongside inscribed angles, problems often feature circumscribed angles — angles between two tangents drawn from a single point outside the circle. A circumscribed angle equals half the difference of the arcs it intercepts.
Worked examples on inscribed and central angles
Example 1. Inscribed angle from an arc
Problem. Inscribed angle \(ABC\) subtends arc \(AC\), equal to \(120°\). Find \(\angle ABC\).
Step 1. Recall the theorem: an inscribed angle equals half the arc it subtends.
Step 2. Substitute: \(\angle ABC = \frac{1}{2} \cdot 120° = 60°\).
Answer: \(60°\).
Example 2. Central angle from an inscribed angle
Problem. Inscribed angle \(ABC\) equals \(40°\). Find central angle \(AOC\), which subtends the same arc \(AC\).
Step 1. A central angle is twice the inscribed angle subtending the same arc: \(\angle AOC = 2 \cdot \angle ABC\).
Step 2. Compute: \(\angle AOC = 2 \cdot 40° = 80°\). Along the way we found the arc: \(AC = 80°\).
Answer: \(80°\).
Example 3. An angle subtending a diameter
Problem. \(AC\) is a diameter of a circle, point \(B\) lies on the circle, \(\angle BAC = 35°\). Find \(\angle BCA\).
Step 1. Angle \(ABC\) subtends diameter \(AC\), so \(\angle ABC = 90°\) — triangle \(ABC\) is right-angled.
Step 2. The sum of a triangle's angles is \(180°\): \(\angle BCA = 180° - 90° - 35° = 55°\).
Answer: \(55°\).
Example 4. A cyclic quadrilateral
Problem. Quadrilateral \(ABCD\) is inscribed in a circle, \(\angle A = 70°\), \(\angle B = 80°\). Find angles \(C\) and \(D\).
Step 1. The sums of opposite angles of a cyclic quadrilateral equal \(180°\): \(\angle C = 180° - \angle A = 180° - 70° = 110°\).
Step 2. Likewise \(\angle D = 180° - \angle B = 180° - 80° = 100°\).
Check. \(70° + 80° + 110° + 100° = 360°\) — the sum of the quadrilateral's angles checks out.
Answer: \(\angle C = 110°\), \(\angle D = 100°\).
Common mistakes with angles in a circle
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Confusing the inscribed angle with the central angle: taking half the arc for the angle whose vertex is at the center (or the other way around).
First look at the vertex. Vertex at the center \(O\) — a central angle, equal to the whole arc. Vertex on the circle — an inscribed angle, equal to half the arc.
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Taking the wrong arc: for an inscribed angle, using the arc where its vertex lies.
An inscribed angle subtends the arc that lies "inside" the angle — the vertex is always on the other arc. In the notation \(\angle ABC\), the angle subtends arc \(AC\) that does not contain point \(B\).
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Multiplying and dividing by 2 in the wrong direction: dividing instead of multiplying when going from an inscribed angle to a central angle.
Remember: the inscribed angle is "small" (half the arc), the central angle is "big" (the whole arc). From inscribed to central — multiply by 2; from central to inscribed — divide.
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Treating any angle whose sides cross the circle as inscribed — for example, an angle between chords that intersect inside the circle.
An inscribed angle must have its vertex on the circle itself. An angle with its vertex inside the circle equals half the sum of the intercepted arcs; with its vertex outside the circle — half the difference: these are different formulas.
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In a cyclic quadrilateral, setting the sum of adjacent angles equal to $180°$ instead of opposite angles.
The property only holds for opposite vertices: \(\angle A + \angle C = 180°\) and \(\angle B + \angle D = 180°\). Adjacent angles of a cyclic quadrilateral can be anything.
Questions and answers
What is an inscribed angle equal to?
An inscribed angle equals half the arc it subtends. For example, if the arc equals 120°, the inscribed angle equals 60°. This is the inscribed angle theorem — the main formula of the topic.
How are the inscribed and central angles subtending the same arc related?
A central angle is twice the inscribed angle: ∠AOC = 2·∠ABC. A central angle equals the arc itself, while an inscribed angle equals half of it.
Why does an inscribed angle subtending a diameter equal 90°?
A diameter subtends a semicircle — an arc of 180°. An inscribed angle equals half the arc: 180° : 2 = 90°. That's why any triangle with one side being a diameter of its circumscribed circle is a right triangle.
Can an inscribed angle be obtuse?
Yes. If the arc subtended by the angle is greater than 180°, the angle is obtuse: for example, an arc of 210° corresponds to an inscribed angle of 105°. An inscribed angle is always less than 180°, though.
What is a circumscribed angle and how does it differ from an inscribed angle?
A circumscribed angle is formed by two tangents drawn to a circle from a single external point. Its vertex lies outside the circle, and it equals half the difference of the intercepted arcs. An inscribed angle has its vertex on the circle and equals half of a single arc.
What other kinds of angles in a circle are there?
Besides central and inscribed angles, there are angles between chords (vertex inside the circle — half the sum of the arcs), angles between secants from an external point (half the difference of the arcs), and angles between a tangent and a chord (half of the arc enclosed inside). All of them are expressed through arcs of the circle.
