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      The angle in a semicircle is a right angle

      Lesson details

      Learning outcome

      I can derive and use the theorem: the angle in a semicircle is a right angle.

      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.

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

      • Diameter - The diameter of a circle is any line segment that starts and ends on the circumference of the circle and passes through the centre of the circle.

      Common misconception

      Pupils may over-generalise the circle theorem by assume that all triangles that are inscribed in a circle are right-angled.

      The triangle is only right-angled if one of its angles is subtended by the diameter. This also means that if the triangle is right-angled, then its hypotenuse is the diameter of the circle.

      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.
      A chord has endpoints of A and B. Where on the circle must points A and B lie?

      At the centre of the circle
      On a tangent to the circle
      Correct answer: On the circumference of the circle
      Inside the circle
      Outside the circle

      Q2.
      The size of the angle labelled $$x$$° on this 20-point geoboard is °.

      An image in a quiz
      Correct Answer: 108, 108°, 108 degrees

      Q3.
      The size of the angle labelled $$y$$° on this 20-point geoboard is °.

      An image in a quiz
      Correct Answer: 36, 36°, 36 degrees

      Q4.
      The size of the angle labelled $$z$$° on this 20-point geoboard is °.

      An image in a quiz
      Correct Answer: 54, 54°, 54 degrees

      Q5.
      What is the size of the angle subtended by the chord AC (in degrees)?

      An image in a quiz
      Correct Answer: 36, 36°, 36 degrees

      Q6.
      The size of the angle labelled $$m$$° is °.

      An image in a quiz
      Correct Answer: 68, 68°, 68 degrees

      6 Questions

      Q1.
      Find the size of angle $$a$$° on this 8-point geoboard (in degrees).

      An image in a quiz
      20°
      Correct answer: 45°
      360°

      Q2.
      Find the size of angle $$b$$° on this 8-point geoboard (in degrees).

      An image in a quiz
      $$180° - 45°$$
      Correct answer: $$\big({{180° - 45°}\over2}\big)$$
      $$45°\over2$$
      Correct answer: 67.5°
      135°

      Q3.
      The size of the angle labelled $$c$$° on this 8-point geoboard is °.

      An image in a quiz
      Correct Answer: 135, 135°, 135 degrees

      Q4.
      The size of the angle $$(b+d)$$° on this 8-point geoboard is °.

      An image in a quiz
      Correct Answer: 90, 90°, 90 degrees

      Q5.
      The size of the angle labelled $$x$$° is °.

      An image in a quiz
      Correct Answer: 76, 76°, 76 degrees

      Q6.
      Chord AB has length of 4 cm. The length of chord BC is cm (2 d.p.).

      An image in a quiz
      Correct Answer: 23.66, 23.66 cm

      To help you plan your 10 maths lesson on: The angle in a semicircle is a right angle, download all teaching resources for free and adapt to suit your pupils' needs...