Circle Geometry
Explore the key theorems of circle geometry including the angle at the centre, cyclic quadrilaterals, and tangent properties.
Angle at the Centre Theorem
One of the most important circle theorems states that the angle at the centre of a circle is twice the angle at the circumference when both angles subtend the same arc.
Angle at Centre Theorem
∠AOB = 2 × ∠ACB
where O is the centre and A, B, C are points on the circle
Angles in the Same Segment
All angles inscribed in the same segment of a circle are equal. If two angles subtend the same chord from the same side, they are equal.
Angle in a Semicircle
An angle inscribed in a semicircle (subtending a diameter) is always 90°. This is a special case of the angle at the centre theorem.
Cyclic Quadrilaterals
A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. These shapes have special angle properties.
Opposite Angles Theorem
Opposite angles in a cyclic quadrilateral are supplementary
∠A + ∠C = 180° and ∠B + ∠D = 180°
Worked Example: Find the missing angle
In cyclic quadrilateral ABCD, ∠A = 75° and ∠B = 110°. Find ∠C and ∠D.
Step 1: Opposite angles are supplementary: ∠C = 180° − ∠A = 180° − 75° = 105°
Step 2: ∠D = 180° − ∠B = 180° − 110° = 70°
Check: 75 + 105 = 180 ✓ 110 + 70 = 180 ✓
The exterior angle of a cyclic quadrilateral equals the interior opposite angle. This is a useful shortcut in harder problems.
Tangent Properties
A tangent to a circle touches it at exactly one point (called the point of tangency). Tangents have two key properties.
Tangent-Radius Property
A tangent to a circle is perpendicular to the radius drawn to the point of tangency. The angle between them is always 90°.
Equal Tangents from External Point
Two tangents drawn from an external point to a circle are equal in length. If PA and PB are tangents from P, then PA = PB.
Worked Example: Tangent from external point
Point P is 13 cm from the centre O of a circle with radius 5 cm. Find the length of the tangent from P to the circle.
Step 1: The tangent PT is perpendicular to the radius OT at the point of tangency.
Step 2: Triangle OTP is right-angled at T: OP = 13, OT = 5
Step 3: By Pythagoras: PT² = OP² − OT² = 169 − 25 = 144
Answer: PT = 12 cm
Chord Properties
Perpendicular Bisector
The perpendicular from the centre to a chord bisects the chord.
Equal Chords
Equal chords are equidistant from the centre of the circle.
Intersecting Chords
If chords AB and CD intersect at P: AP × PB = CP × PD
Key Vocabulary
Subtend
An angle is said to subtend an arc when the two sides of the angle meet the endpoints of that arc.
Cyclic Quadrilateral
A quadrilateral whose four vertices all lie on the circumference of a single circle.
Tangent
A straight line that touches a circle at exactly one point and is perpendicular to the radius at that point.
Supplementary
Two angles are supplementary when they add to 180°. Opposite angles in a cyclic quadrilateral are supplementary.
Knowledge Check
Test your understanding of circle geometry theorems.
Question 1
An angle at the centre of a circle is 140°. What is the inscribed angle subtending the same arc?
Question 2
In a cyclic quadrilateral, one angle is 65°. What is the opposite angle?
Question 3
A tangent meets a radius at the point of tangency. What is the angle between them?
Question 4
An angle inscribed in a semicircle always equals:
Question 5
Two tangents are drawn from external point P to a circle. If one tangent segment is 9 cm, how long is the other?
Key Concepts Summary
- ●The angle at the centre is twice the inscribed angle subtending the same arc.
- ●Angles in the same segment are equal; an angle in a semicircle is always 90°.
- ●Cyclic quadrilateral: opposite angles are supplementary (add to 180°).
- ●A tangent is perpendicular to the radius at the point of tangency.
- ●Equal tangents: two tangents from an external point to a circle are equal in length.