Circle properties
Circle geometry connects angles and lengths through stated facts: equal radii, a tangent at a named point, or points on the circumference. Name the relevant chord or arc before applying an angle theorem.
What this guide covers
Chord and tangent properties, angles subtended by the same arc, semicircles and cyclic quadrilaterals. Conditions matter more than how a sketch looks. The graph also contains an alternate-segment objective, which is not included here because it is not explicitly listed in this syllabus scope.
Syllabus reference: 2026 Mathematics 4052, G3, page 9. These are Mentora search groupings, not new official syllabus categories.
The content in this guide was also checked against 2027 G3 Mathematics K310, G3, page 9 on 2026-09-12. This is subject-content alignment, not a claim of complete syllabus coverage.
Worked example
O is the centre. A, B and C lie on the circle, with C on the major arc AB. The minor central angle AOB is 124°. Find angle ACB.
Both angles stand on the minor arc AB. The angle at the centre is twice the angle at the circumference, so angle ACB = 124° ÷ 2 = 62°. The position of C ensures that the intended arc is the same.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
AB is a diameter and C is a different point on the circle. Find angle ACB.
Show worked answer
An angle subtended by a diameter at the circumference is a right angle, so angle ACB = 90°.
Check 2
A, B, C, D are in order on a circle. Angle ABC is 107°. Find angle ADC.
Show worked answer
Opposite angles of a cyclic quadrilateral sum to 180°, so angle ADC = 180° − 107° = 73°.
Check 3
PT is tangent at T to a circle with centre O. OP = 13 cm and OT = 5 cm. Find PT.
Show worked answer
A radius is perpendicular to the tangent at the point of contact, so triangle OTP is right-angled at T. PT² = 13² − 5² = 144, giving PT = 12 cm.
Common mistakes to watch for
- Do not halve just any central angle: identify the shared arc and check which side of the chord the point lies on.
- A tangent is perpendicular to the radius only at its point of contact.
- Opposite angles sum to 180° only when the quadrilateral is cyclic; a sketch alone does not establish this.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
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More G3 Math topicsPublished by Mentora · Sources checked 12 Sept 2026
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Original public examples and answers were checked with automated mathematical/editorial QA. No individual author, human reviewer or independent expert review is recorded.