Circle Theorems — Definition, Formula & Examples
Circle theorems are a set of rules that describe relationships between angles, chords, tangents, and arcs formed in and around a circle. They let you find unknown angles and lengths without measuring, using logical reasoning instead.
Circle theorems are geometric propositions proved from Euclid's axioms that govern the measure of inscribed angles, central angles, angles in semicircles, angles in the same segment, opposite angles of cyclic quadrilaterals, and the properties of tangent lines to a circle.
How It Works
Each theorem gives you a shortcut for a specific configuration. The **inscribed angle theorem** says any inscribed angle is half the central angle that subtends the same arc. The **angle in a semicircle** theorem says an angle inscribed in a semicircle is always . The **same segment theorem** says inscribed angles subtending the same arc are equal. The **cyclic quadrilateral theorem** says opposite angles in a quadrilateral inscribed in a circle sum to . The **alternate segment theorem** says the angle between a tangent and a chord equals the inscribed angle in the alternate segment. When solving problems, identify which configuration matches your diagram, then apply the matching theorem.
Worked Example
Problem: A circle has centre O. Points A, B, and C lie on the circle. The central angle AOB is 130°. Find the inscribed angle ACB.
Identify the theorem: Angle ACB is an inscribed angle and angle AOB is the central angle. Both subtend the same arc AB. By the inscribed angle theorem, the inscribed angle is half the central angle.
Apply the theorem: Divide the central angle by 2.
Answer: The inscribed angle ACB is .
Why It Matters
Circle theorems appear throughout GCSE and high-school geometry exams and are essential for proof-writing. They also underpin real-world applications such as designing circular gears, satellite dish reflectors, and any engineering scenario where arcs and tangent lines must be precisely calculated.
Common Mistakes
Mistake: Confusing which angle is the inscribed angle and which is the central angle, then dividing or doubling the wrong one.
Correction: The central angle has its vertex at the centre of the circle; the inscribed angle has its vertex on the circle. The inscribed angle is always half the central angle for the same arc.