Circle Theorems
Circle theorems are a set of geometric rules that describe angle and chord relationships within and around circles.
What You Need to Know
Key Concept Diagram
The angle at the centre is twice the angle at the circumference when both subtend the same arc
Angles in the same segment subtending the same arc are equal
The angle in a semicircle (subtended by a diameter) is always 90°
Opposite angles in a cyclic quadrilateral are supplementary (sum to 180°)
Key Vocabulary
Cyclic Quadrilateral
A quadrilateral whose four vertices all lie on the circumference of a circle
Subtend
An angle or arc subtends at a point when the lines from that point to the endpoints of the arc form the angle
Chord
A straight line segment joining two points on the circumference of a circle
Tangent-Chord Angle
The angle between a tangent to a circle and a chord drawn from the point of tangency, equal to the inscribed angle on the opposite side
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
The angle at the centre of a circle is 110°. What is the angle at the circumference subtending the same arc?
Question 2
A chord is a diameter. What is the angle in the semicircle?
Question 3
In a cyclic quadrilateral, one angle is 75°. What is the opposite angle?
Key Concepts Summary
- ●The angle at the centre is twice the angle at the circumference when both subtend the same arc
- ●Angles in the same segment subtending the same arc are equal
- ●The angle in a semicircle (subtended by a diameter) is always 90°
- ●Opposite angles in a cyclic quadrilateral are supplementary (sum to 180°)