Cambridge IGCSE·Mathematics 0580·Extended Circle Theorems

📋Key knowledge — Geometry

Formulas & equations

Angles in a triangle: sum = 180°
Angles in a quadrilateral: sum = 360°
Exterior angle of a triangle = sum of the two non-adjacent interior angles
Interior angle of regular n-gon = (n−2) × 180° ÷ n
Exterior angle of regular n-gon = 360° ÷ n
Arc length = (θ/360) × 2πr
Area of sector = (θ/360) × πr²

Key facts & rules

Corresponding angles are equal (parallel lines, F-shape).
Alternate angles are equal (parallel lines, Z-shape).
Co-interior (allied) angles add up to 180° (parallel lines, C-shape).
Tangent is perpendicular to radius at point of contact.
Angle in semicircle = 90° (angle in a semicircle theorem).
Angle at centre = 2 × angle at circumference (same arc).
Angles in the same segment are equal.
Opposite angles in a cyclic quadrilateral add to 180°.
Similar shapes: all angles equal, sides in the same ratio. Area ratio = k², volume ratio = k³.

Laws & theorems

Pythagoras: a² + b² = c² (right-angled triangles only).
Congruence conditions: SSS, SAS, ASA, RHS.

IGCSE Geometry — Circle theorems Practice Extended

7 Cambridge IGCSE 0580 questions on circle theorems at the Extended tier — each with its full mark scheme. For the Core tier on the same skill, click here.

How to approach circle theorems — open the study guide

Circle theorems on IGCSE 0580

Key theorems: angle in semicircle = 90°; angle at centre = 2 × angle at circumference; angles in same segment are equal; opposite angles in cyclic quadrilateral sum to 180°; tangent is perpendicular to radius; alternate segment theorem.