What are circle theorems?

• You will have learned a lot of angle facts for your GCSE, including angles in polygons and angles with parallel lines
• Circle Theorems deal with angle facts that occur when lines are drawn within and connected to a circle

What do I need to know?

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

• 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, so ensure you’re familiar with them
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    Angles at Centre & Circumference

    Circle Theorem: The angle subtended by an arc at the centre is twice the angle at the circumference

    • This is one of the most useful circle theorems and forms a basis for many other angle facts within circles
    • In this theorem, the chords (radii) to the centre and the chords to the circumference are both drawn from (subtended by) the ends of the same arc
    • It is an easy circle theorem to spot on a diagram
      STEP 1
      Find any two radii in the circle and follow them to the circumference
      STEP 2
      See if there are lines from those points going to any other point on the circumference
    • If you are asked for a reason in an exam and you use this theorem, use the key phrase;
      • “The angle at the centre is twice the angle at the circumference”

    

    • This theorem can also happen when the ‘triangle parts’ overlap:

    

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

    • This is a special case of the angle at the centre theorem above
      • The angle on the diameter = 180°
      • The angle at the circumference = 90°
    • It is easy to spot, look for a diameter in the circle and see if it makes the base of a triangle, with its top vertex at the circumference
      • Make sure that you are looking at a diameter by checking it goes through the centre
      • These questions only need half of the circle so they could appear in whole circles or in semicircles only
    • Any angle at the circumference that comes from each end of the diameter in this way will be 90°
    • This is most commonly known as the angle in a semicircle theorem, however if you are asked for a  reason in an exam and you use this theorem, use the key phrase; 
      • “The angle in a semicircle is 90°”  
    • Look out for triangles hidden among other lines/shapes within the circle