4.7 Circle Theorems I

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  • Calculate unknown angles using the property that the angle in a semicircle = 90°
  • Calculate unknown angles using the property that the angle between a tangent and a radius = 90°
  • Calculate unknown angles using the property that the angle at the centre is twice the angle at the circumference
  • Calculate unknown angles using the property that angles in the same segment are equal
  • Calculate unknown angles using the property that opposite angles of a cyclic quadrilateral sum to 180°
  • Apply the alternate segment theorem to find unknown angles
  • Give clear written reasons for every answer you calculate, using the correct geometric language

Parts of a Circle — Know These First

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Before diving into the theorems, you need to know the key parts of a circle. These words will appear constantly in exam questions.

  • Centre — the point exactly in the middle of the circle
  • Radius (plural: radii) — a straight line from the centre to any point on the circumference. All radii in the same circle are equal in length.
  • Diameter — a straight line that passes through the centre and touches the circumference at both ends. It is the longest possible chord. The diameter is exactly twice the radius.
  • Circumference — the curved outer boundary (edge) of the circle
  • Chord — any straight line that connects two points on the circumference. A diameter is a special chord that passes through the centre.
  • Arc — a portion (piece) of the circumference between two points
  • Tangent — a straight line that touches the circle at exactly one point and does not cross into the circle
  • Segment — the region between a chord and the arc it cuts off. The major segment is the larger region; the minor segment is the smaller region.
  • Sector — the region between two radii and the arc connecting them (shaped like a pizza slice)

💡 Important habit: Every time you look at a circle diagram, start by marking all the radii. Because all radii are equal, they create isosceles triangles — and knowing that will help you find angles.

The Circle Theorems

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Theorem 1 — Angle in a Semicircle = 90°

The rule: If you draw a triangle inside a circle so that one side of the triangle is the diameter, the angle at the point on the circumference (opposite the diameter) is always exactly 90°.

Why does this happen? A diameter divides a circle into two equal halves — two semicircles. Any angle drawn from the two ends of the diameter up to the circumference will always land on the curve of one of those semicircles, giving exactly 90°.

How to spot it in a diagram:

  • Look for a triangle where one side is a diameter (it passes through the centre)
  • All three corners of the triangle must be on the circumference
  • The 90° angle is always at the corner opposite the diameter

What to write in an exam: "The angle in a semicircle is 90°"

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