Angles at Centre & Semicircles (OCR GCSE Maths)

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
  •   (C = πd)   
  •     (A = πr²)

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