Angle in a Semicircle
Simple Explanation
Any angle inscribed in a semicircle — with its vertex on the circle and the two chords reaching to the ends of a diameter — is always exactly 90°.
Why Do We Need It?
This is a fast, reliable way to construct or check a right angle using only a circle and a straightedge, and it comes up constantly in circle-geometry proofs.
See It
A circle with a diameter AB and a point C on the circle, forming a right angle at C
Formula
The Inscribed Angle Theorem
Inscribed angle = (1/2) × 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
- — the angle at the centre of the circle, formed by two radii to the ends of the arc
- inscribed angle
- — 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
Use the semicircle angle to find a missing angle
In a semicircle, ∠ACB = 90° with C on the circle. If ∠CAB = 35°, find ∠CBA.
Why Does This Work?
This is a direct corollary of the inscribed angle theorem: a diameter is itself an arc of 180° as seen from the centre (a straight line through it), so the central angle for that arc is 180° — making the inscribed angle exactly half of that, 90°, no matter where C sits on the semicircle.
Real-Life Example
Thales' theorem in carpentry
A carpenter needs to find a point that forms a perfect right angle relative to two fixed points (the ends of a beam).
Drawing a circle with the beam as diameter guarantees any point marked on that circle forms a 90° angle with the beam's ends — a practical use of the semicircle angle rule.
Practice
In a semicircle with diameter AB, C is on the circle. If ∠ABC = 62°, find ∠BAC.
MediumCommon mistake
Applying the 90° rule to any triangle inscribed in a circle — it only holds when one side of the triangle is specifically a diameter, not for an arbitrary chord.
Quick Review
- Any angle inscribed in a semicircle (subtending a diameter) is exactly 90°.
- This is the special case of the inscribed angle theorem where the central angle is 180°.
- A fast way to construct or verify a right angle using a circle.