Circle theorems
Section: Geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Angle at the Centre and Angle in a Semicircle
Extended Only
Circle theorems describe fixed relationships between angles formed by chords, radii and tangents. The first key theorem relates an angle at the centre to an angle at the circumference standing on the same arc.
- The angle at the centre of a circle is always twice the angle at the circumference, when both angles stand on the same arc
- The angle in a semicircle is always 90° - this is a special case of the centre-circumference rule, since the angle at the centre for a diameter is 180° (a straight line)
- The diameter of a circle always subtends a right angle at any point on the circumference
Worked Example: Using the Centre-Circumference Angle Rule
- Question: Points A, B and C lie on a circle with centre O. The angle at the circumference, angle ABC, is 42°. Find the angle at the centre, angle AOC, standing on the same arc AC.
- Step 1: The angle at the centre is twice the angle at the circumference
- Step 2: 242=84
- Answer: angle AOC = 84°
Worked Example: Using the Angle in a Semicircle
- Question: AB is a diameter of a circle, and C is a point on the circumference. Angle BAC = 27°. Find angle ACB and angle ABC.
- Step 1: AB is a diameter, so angle ACB (the angle in the semicircle) = 90°
- Step 2: Angles in triangle ABC sum to 180°: 180-90-27=63
- Answer: angle ACB = 90°, angle ABC = 63°
Common Mistakes
MistakeDividing the circumference angle by 2 to find the centre angle, instead of multiplying
Fixthe angle at the centre is always larger - multiply the circumference angle by 2, don't divide it
MistakeAssuming the angle in a semicircle rule applies to any chord, not just a diameter
Fixthe 90° rule only applies when the chord is specifically a diameter (passing through the centre) - any other chord gives a different angle
Angles in the Same Segment and Cyclic Quadrilaterals
Extended Only
Angles standing on the same arc from the same side are always equal, and this leads directly to a useful rule about four-sided shapes drawn inside a circle.
- Angles in the same segment (standing on the same arc, from the same side of it) are always equal
- A cyclic quadrilateral is a four-sided shape with all four vertices on the circumference of a circle
- Opposite angles of a cyclic quadrilateral always sum to 180°
Worked Example: Using Angles in the Same Segment
- Question: Points P, Q, R and S lie on a circle. Angle PQS = 48°, and angle PQS and angle PRS both stand on the same arc PS, from the same side. Find angle PRS.
- Step 1: Angles in the same segment are equal
- Answer: angle PRS = 48°
Worked Example: Using the Cyclic Quadrilateral Rule
- Question: WXYZ is a cyclic quadrilateral. Angle WXY = 98°. Find angle WZY.
- Step 1: Opposite angles of a cyclic quadrilateral sum to 180°
- Step 2: 180-98=82
- Answer: angle WZY = 82°
Common Mistakes
MistakeAssuming all angles in a cyclic quadrilateral are related, rather than specifically opposite pairs
Fixonly opposite angles (not adjacent ones) are supplementary in a cyclic quadrilateral
MistakeConfusing "angles in the same segment" with "angle at the centre", and doubling or halving unnecessarily
Fixangles in the same segment (both at the circumference) are simply equal - no doubling or halving is needed, unlike the centre-circumference rule
Tangents and the Alternate Segment Theorem
Extended Only
A tangent touches a circle at exactly one point, and has its own special angle relationships with the radius at that point and with chords drawn from it.
- A tangent to a circle is always perpendicular (at 90°) to the radius drawn to the point of contact
- The alternate segment theorem: the angle between a tangent and a chord drawn from the point of contact equals the angle subtended by that chord in the alternate segment (the segment on the other side of the chord)
- Tangents drawn from the same external point to a circle are equal in length
Worked Example: Using the Tangent-Radius Property
- Question: A tangent touches a circle at point T. The radius OT is drawn to the same point. Find the angle between the tangent and the radius OT.
- Step 1: A tangent is always perpendicular to the radius at the point of contact
- Answer: 90°
Worked Example: Using the Alternate Segment Theorem
- Question: A tangent touches a circle at point E. A chord EF is drawn from E, and the angle between the tangent and EF is 52°. Find the angle subtended by EF at a point G on the circumference, in the alternate segment.
- Step 1: By the alternate segment theorem, the tangent-chord angle equals the inscribed angle in the alternate segment
- Answer: angle EGF = 52°
Common Mistakes
MistakeAssuming the alternate segment theorem angle is found by subtracting from 90° or 180°, rather than simply being equal
Fixthe tangent-chord angle and the alternate segment angle are exactly equal - no further calculation needed once the correct pair is identified
MistakeApplying the alternate segment theorem to the "near" segment (the same side as the marked angle) instead of the "far" (alternate) segment
Fixthe equal angle is always in the segment on the other side of the chord from the tangent angle - hence "alternate"
Interactive revision notes, videos and practice questions load below.