The inscribed angle and its relationship to the central angle

Definition of an inscribed angle

An inscribed angle is an angle whose vertex is a point on the circle itself (not the centre), and whose two sides are chords passing through two other points on the circle.

Like the central angle, the inscribed angle subtends an arc of the circle.

The fundamental property

Here is the most important rule in this chapter:

An inscribed angle measures half the central angle that subtends the same arc.

In other words, if the central angle is a, then any inscribed angle subtending the same arc is a / 2.

Example

If a central angle subtends an arc and measures 80 degrees, then any inscribed angle subtending that same arc measures 80 / 2 = 40 degrees.

Another way of looking at the same relationship: if an inscribed angle measures 35 degrees, the corresponding central angle measures 2 × 35 = 70 degrees.

Consequence: the special case of a semicircle

If the intercepted arc is a semicircle, the corresponding central angle is 180 degrees (this is a straight angle). The inscribed angle is therefore 180 / 2 = 90 degrees.

Important consequence: a triangle inscribed in a circle, one side of which is a diameter, is always a right-angled triangle (the vertex opposite the diameter forms a right angle). This is a property that is widely used in geometry.

Summary table

Central angle	Inscribed angle (same arc)
180 degrees	90 degrees
100 degrees	50 degrees
60 degrees	30 degrees

Pitfalls to avoid

• Check carefully that the central angle and the inscribed angle do indeed subtend the same arc before applying the relationship.
• Do not reverse the relationship: it is the inscribed angle that is half the central angle, and not the other way round.
• Two inscribed angles that subtend the same arc always have the same measure between them, even if their vertices lie at different points on the circle.