Angle in a Semicircle (Edexcel IGCSE Maths)

Spec point — 4.6C

Angle in a semicircle

Circle Theorem: The angle in a semicircle is 90°

• The lines drawn from a point on the circumference to either end of a diameter are perpendicular

  • The angle at that point on the circumference is 90°

  • This circle theorem only uses half of the circle

    • The right-angle is called the angle in a semicircle

• This is a special case of the circle theorem "the angle at the centre is twice the angle at the circumference"

  • The angle on the diameter is 180°
  • The angle at the circumference is halved, giving 90°

• To spot this circle theorem on a diagram look for a triangle where

  • one side is the diameter

    • Remember that a diameter always goes through the centre

  • all three vertices are on the circumference

• The 90º angle will always be the angle opposite the diameter

• When explaining this theorem in an exam you must use the keywords:

  • The angle in a semicircle is 90°

• Questions that use this theorem may

  • appear in whole circles or in semicircles
  • require the use of Pythagoras' Theorem to find a missing length

Worked Example

P, Q and R are points on a circle. RQ is a diameter.

Find the value of $y$.

Give a reason for your answer.

Answer:

Use the fact that angles in a triangle add up to 180º and the circle theorem

The angle in a semicircle is 90°

Write an equation for $y$

$y+90+40=180$

Solve for $y$

$y=180-90-40\,y=50$

$y=50$

Final answer: The angle in a semicircle is 90°