Central and Inscribed Angles
Simple Explanation
A central angle has its vertex at the centre of the circle; an inscribed angle has its vertex on the circle itself. The Inscribed Angle Theorem states that when both angles look at (subtend) the exact same arc, the inscribed angle is always exactly half the central angle.
Why Do We Need It?
This relationship connects two very different-looking angles through the same arc β the foundation for almost every other circle-angle theorem in this chapter.
See It
A circle with a central angle at O and an inscribed angle at C, both looking at the same arc AB, with the inscribed angle exactly half the central angle
Formula
The Inscribed Angle Theorem
Inscribed angle = (1/2) Γ central angle, for angles subtending the same arc
An angle inscribed in a circle (with its vertex on the circle) is always exactly half the central angle that subtends the same arc.
- central angle
- β the angle at the centre of the circle, formed by two radii to the ends of the arc
- inscribed angle
- β the angle at a point on the circle, formed by two chords to the same ends of the arc
When to use it: Whenever an angle is formed at the centre and another at the circle's edge, both looking at the same arc.
Worked Example
Find an inscribed angle from a central angle
A central angle subtending arc AB measures 80Β°. Find the inscribed angle at point C on the major arc, subtending the same arc AB.
Why Does This Work?
Stated simply: draw the radius OC and extend it. In the resulting isosceles triangles (each formed by two radii), base angles are equal β using the exterior angle theorem (an exterior angle of a triangle equals the sum of the two remote interior angles) on each of these isosceles triangles shows that the central angle splits into two parts, each exactly double the corresponding part of the inscribed angle, giving central = 2 Γ inscribed overall.
Real-Life Example
Camera field-of-view geometry
Photographers and architects use circle geometry to reason about sightlines and viewing angles from different positions around a curved space.
The inscribed angle theorem explains why viewers at different points around a circular room see the same wall segment under a predictably related angle, depending on where the "central" reference point is.
Practice
An inscribed angle measures 35Β°. Find the central angle subtending the same arc.
MediumCommon mistake
Forgetting that both angles must subtend the exact same arc β an inscribed angle and a central angle looking at different arcs of the same circle have no such fixed relationship.
Quick Review
- Inscribed angle = (1/2) Γ central angle, when both subtend the same arc.
- Central angle: vertex at the centre. Inscribed angle: vertex on the circle.
- Proof idea: uses the exterior angle theorem on isosceles triangles formed by radii.