IGCSEMaths: CoreCIERevision NotesGeometryCircle TheoremsAngle in a Semicircle

Syllabus Edition

First teaching 2023

First exams 2025

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Angle in a Semicircle (CIE IGCSE Maths: Core)

Revision Note

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Jamie W

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Jamie W

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Maths

Circle Theorems

What are circle theorems?

  • Circle Theorems deal with angle facts that occur when lines are drawn within, or connected to, a circle
  • These can be used in conjunction with the other angle facts you have learned, to solve geometry problems and find missing angles
  • To solve some problems you may need to use the angle facts you are already familiar with from triangles, polygons, and parallel lines
  • You may also have to use the formulae for circumference and area
    • Circumference space equals straight pi cross times diameter
    • Area space equals space straight pi straight r squared

What do I need to know about circles?

  • You must be familiar with the names of parts of a circle including radius, diameter, arc, sector, chord, segment and tangent

Circle terminology

Angle in a Semicircle

Circle theorem: The angle in a semicircle is a right angle

  • Suppose you have a triangle in a circle where
    • two vertices at either end of a diameter
    • the third vertex is on the circumference of the circle
  • Then the angle at the third vertex will be a right angle (90°)
  • This circle theorem may be seen in a full circle with a diameter drawn, or in a semicircle
  • Look for a diameter in the circle and see if it forms the hypotenuse of a triangle, with the opposite vertex at the circumference
    • Remember that a diameter must pass through the centre of the circle
  • Because this theorem involves a right-angled triangle, some questions may require use of Pythagoras' Theorem to find one of the lengths if the other two are known
  • This is most commonly known as the angle in a semicircle theorem
  • However if using it in an exam you must use the keywords 
    • The angle in a semicircle is 90°

Right angle in a semi-circle circle theorem

Exam Tip

  • As soon as you spot this arrangement in a question, mark the angle as 90 degrees on the diagram
    • Sometimes just doing this will earn you a mark!

Worked example

A, B, and C are points on a circle. AC is a diameter of the circle. Find the value of a.

angle in semicircle worked example question diagram

As the line AC is the diameter of the circle, we can use the circle theorem "the angle in a semicircle is a right angle" to state that angle B must be 90 degrees

We can now use the fact that the internal angles of a triangle sum to 180, to find the unknown angle

a + 53 + 90 = 180

a + 143 = 180

a = 37

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    Jamie W

    Author: Jamie W

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.