Circle Theorems
Circle theorems describe relationships between angles, chords, and tangents in and around circles, enabling geometric proofs and problem solving.
What You Need to Know
Key Concept Diagram
The angle at the centre theorem: the central angle is twice any inscribed angle subtended by the same arc
Angles in the same segment (subtended by the same chord) are equal
The angle in a semicircle is always 90° — a right angle
The tangent to a circle is perpendicular to the radius drawn to the point of tangency
Key Vocabulary
Inscribed angle
An angle formed by two chords that share an endpoint on the circle
Central angle
An angle whose vertex is the centre of the circle and whose sides are radii
Chord
A straight line segment joining two points on the circumference of a circle
Tangent
A line that touches a circle at exactly one point and is perpendicular to the radius at that point
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
An inscribed angle subtends an arc. The central angle subtending the same arc is 80°. What is the inscribed angle?
Question 2
A triangle is inscribed in a semicircle with its hypotenuse as the diameter. What is the angle at the circumference opposite the diameter?
Question 3
Two tangents are drawn from an external point P to a circle. Which of the following is true?
Key Concepts Summary
- ●The angle at the centre theorem: the central angle is twice any inscribed angle subtended by the same arc
- ●Angles in the same segment (subtended by the same chord) are equal
- ●The angle in a semicircle is always 90° — a right angle
- ●The tangent to a circle is perpendicular to the radius drawn to the point of tangency