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      The angle at the centre of the circle is twice the angle at any point on the circumference

      Lesson details

      Learning outcome

      I can derive and use the theorem: the angle at the centre of the circle is twice the angle at any point on the circumference.

      Key learning points

      1. A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments
      2. Theorems can be thought of as puzzles to solve, you are showing how to find a result
      3. In order to use this theorem, you may need to draw a diagram or add more information to an existing one

      Keywords

      • Subtended - An angle can be subtended by a line segment or curve. The legs of the subtended angle meet the endpoints of the line segment or curve.

      • Arc - An arc of a circle is part of the circle’s circumference.

      • Radius - The radius is any line segment that joins the centre of a circle to any point on its circumference.

      • Chord - A chord is any line segment joining two points on the circumference of a circle.

      Common misconception

      When faced with a quadrilateral which has three vertices on a circle and one vertex at the centre, pupils may think that the interior angle at the centre is twice the angle at the circumference.

      The two angles need to be subtended by the same arc for this circle theorem to apply. Therefore, it is the reflex angle at the centre of the circle which is twice the angle at the circumference.

      Teacher tip

      The lesson demonstrates how this circle theorem could be explored practically by using a circular geoboard and using dynamic geometry software. Pupils may wish to explore these further for themselves.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      What is the dashed line segment on this circle called?

      An image in a quiz
      arc
      chord
      sector
      segment
      Correct answer: tangent

      Q2.
      The line segment on this circle passes through its centre. What is the shaded region on this circle called?

      An image in a quiz
      Correct answer: sector
      Correct answer: segment
      chord
      Correct answer: semicircle
      arc

      Q3.
      Which of these statements are correct?

      A radius is needed to form a segment.
      A diameter is needed to form a segment.
      Correct answer: A chord is needed to form a segment.
      Correct answer: An arc is needed to form a segment.
      The circumference forms part of a segment.

      Q4.
      The size of the angle subtended by the chord AB is °.

      An image in a quiz
      Correct Answer: 40, 40°, 40degrees, 40 degrees

      Q5.
      Match each circle to their area (3 s.f.)

      Correct Answer:area = 78.5 cm²,radius = 5 cm
      Correct Answer:area = 314 cm²,diameter = 20 cm
      Correct Answer:area = 1260 cm²,circumference = 40π cm
      Correct Answer:area = 201 cm²,“circle with diameter 8 more than its radius”
      Correct Answer:area = 452 cm²,“circle with radius 12 less than its diameter”

      Q6.
      A 36° angle at the centre (O) of a circle has been subtended by the arc XY. The circle has a radius of 50 cm. The area of the shape bound the arc XY and the legs of the 36° angle is cm² (2 s.f.).

      Correct Answer: 790, 790 cm², 790cm²

      6 Questions

      Q1.
      The size of angle $$x$$° on this 12-point geoboard is °.

      An image in a quiz
      Correct Answer: 30, 30°, 30 degrees, 30degrees

      Q2.
      The size of angle $$y$$° on this 12-point geoboard is °.

      An image in a quiz
      Correct Answer: 75, 75°, 75 degrees, 75degrees

      Q3.
      Angle $$a$$° is subtended by chord TU on this 12-point geoboard. $$a$$° = °

      An image in a quiz
      Correct Answer: 150, 150°, 150 degrees, 150degrees

      Q4.
      Angle $$b$$° is subtended by chord TU at the circumference of this 12-point geoboard. $$b$$° = °

      An image in a quiz
      Correct Answer: 75, 75°, 75 degrees, 75degrees

      Q5.
      The size of angle $$c$$° is °.

      An image in a quiz
      Correct Answer: 52, 52°, 52 degrees, 52degrees

      Q6.
      The size of the angle marked $$d$$° is °.

      An image in a quiz
      Correct Answer: 114, 114°, 114 degrees, 114degrees

      To help you plan your 10 maths lesson on: The angle at the centre of the circle is twice the angle at any point on the circumference, download all teaching resources for free and adapt to suit your pupils' needs...