Circle Theorems

Circles hide a surprising number of consistent angle rules once you know where to look. This topic covers the key circle theorems used throughout ZIMSEC O-level geometry.

  • Angles at the centre and circumference
  • Angles in a semicircle and cyclic quadrilaterals
  • Tangent properties

Lessons for Circle Theorems

Angle at the Centre and Circumference

The centre always "sees" moreTheorem: The angle at the centre of a circle is twice the angle at the circumference, …

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Angles in a Semicircle and Cyclic Quadrilaterals

Angle in a semicircleTheorem: Any angle in a semicircle (subtended by the diameter) is exactly 90° - always, no matter …

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Tangent Properties

Where a straight line just touches the circleTheorem 1: A tangent is always perpendicular (90°) to the radius at the …

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Exercises for Circle Theorems

Centre and circumference angles

The angle at the circumference is 42°. Find the angle at the centre.The angle at the centre is 130°. Find …

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Semicircle and cyclic quadrilateral angles

A triangle is inscribed in a semicircle with the hypotenuse as diameter. State the size of the angle opposite the …

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Tangent problems

A tangent meets a radius at the point of contact. State the angle between them.Two tangents from an external point …

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