Bringing the circle facts together
This topic combines three families of circle facts: circumference and central angles, cyclic quadrilaterals, and tangents. Real problems often need two of them at once.
Key idea
Central angle = 2 × angle at the circumference (same arc); opposite angles of a cyclic quadrilateral sum to 180°; a tangent is perpendicular to the radius at the point of contact.
Angles from arcs
If the central angle on an arc is 160°, the angle at the circumference on that arc is 160 ÷ 2 = 80°. Angles at the circumference standing on the same arc are all equal, so two such angles of 37° each stay 37°.
Worked example
A tangent PA touches a circle, centre O, radius 9 cm. The distance OP = 15 cm. Since OA ⊥ PA, PA = √(15² − 9²) = √(225 − 81) = √144 = 12 cm. If a cyclic quadrilateral in the same figure has one angle 108°, its opposite angle = 180 − 108 = 72°.
Choosing the right rule
Ask: does the angle sit at the centre or the edge? Are four points on the circle? Is a line touching the circle? Matching the picture to the rule is the key skill, and many exam questions hide two rules in one diagram.
A good habit is to mark every known angle and every right angle on the figure before calculating. Once the diagram is fully labelled, the missing angle or length usually comes from a single halving, a subtraction from 180°, or one use of Pythagoras.
Remember
- Halve a central angle to get the circumference angle.
- Cyclic quadrilateral: opposite angles add to 180°.
- Tangent ⊥ radius makes a right-angled triangle.