Key ideas
- The angle in a semicircle is 90°. If a triangle is drawn from the two ends of a diameter to any point on the circle, the angle at that point is a right angle.
- A tangent meets a radius at 90° (where the tangent touches the circle).
- Two radii are the same length, so a triangle with two radii as sides is isosceles: its two base angles are equal.
- Then use the facts you already know: angles in a triangle add to 180°, angles on a straight line add to 180°.
- In an exam, mark the angles on the diagram as you find them and give a reason for each one. Often the diagram does most of the work: a right-angle box, then the angles in the triangle.
- When you spot two radii, draw a small dash across each one to show they are equal. That reminds you the triangle is isosceles.
- In the worked examples and answers, angles in green are ones we have worked out and written on the diagram.
Angle in a semicircle
AB is a diameter of the circle, centre O. C is a point on the circumference. Angle CAB = 34°. Calculate the size of angle x.(2 marks)
Spot the semicircle.AB is a diameter, so angle ACB = 90°✓1(angle in a semicircle)
On the diagram
In plain EnglishThe triangle is drawn from both ends of a diameter to a point on the circle. That always makes a right angle at the point on the circle, C.
Use the angles in a triangle.x = 180° − 90° − 34°x = 56°✓2
On the diagram
In plain EnglishThe three angles in triangle ABC add to 180°. Once the right angle is marked on the diagram, you only need the third angle.
⚠ The right angle isn't marked. In the exam it won't be. You have to spot the diameter and know the angle is 90°.
Tangent and radius
PT is a tangent to the circle, centre O, touching it at T. Angle TPO = 28°. Calculate the size of angle x.(2 marks)
Spot the tangent and the radius.OT is a radius and PT is a tangent, so angle OTP = 90°✓1
On the diagram
In plain EnglishA tangent always meets the radius at a right angle, at the point where it touches the circle.
Use the angles in a triangle.x = 180° − 90° − 28°x = 62°✓2
On the diagram
In plain EnglishThe three angles in triangle OTP add to 180°.
Isosceles triangles
A and B are points on the circle, centre O. Angle AOB = 110°. Calculate the size of angle x.(2 marks)
Spot the two radii.OA = OB (radii), so triangle OAB is isosceles✓1
On the diagram
In plain EnglishOA and OB both go from the centre to the circle, so they are the same length. Draw a small dash across each radius to show this. That makes the angles at A and B equal.
Share out what's left.180° − 110° = 70°x = 70°2 = 35°✓2
On the diagram
In plain EnglishTake the 110° away from 180°, then split the rest equally between the two base angles.
Putting it together
The line PQ is a tangent to the circle, centre O, at A. AB is a chord. Angle PAB = 55°. Calculate the size of angle x.(3 marks)
Tangent and radius.Angle OAP = 90° (tangent and radius)Angle OAB = 90° − 55° = 35°✓1
On the diagram
In plain EnglishOA is a radius and PQ is a tangent at A, so they meet at 90°. The chord AB splits that 90° into 55° and the angle we want, OAB.
Isosceles triangle.OA = OB (radii), so angle OBA = 35°✓2
On the diagram
In plain EnglishTriangle OAB has two radii as sides, so mark them equal with a dash and copy the 35° across to the other base angle.
Angles in a triangle.x = 180° − 35° − 35°x = 110°✓3
On the diagram
In plain EnglishThe angles in triangle OAB add to 180°.
Tip. Write each angle on the diagram as you find it. Each one often unlocks the next.
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Your turn
Non-calculator. Type angles as numbers (you can leave out the ° sign). Diagrams are not drawn to scale. Stuck? Tap Need a hint?
Past paper questions
Ready for exam-style questions? These collect National 5 past paper questions on angles, including angles in circles, with answers.
Angles in shapes (includes circles)Every National 5 angles question in one PDF, with answers. The circle questions use the same facts as this page.maths.scot · PDF Open ↗ (opens in a new tab) Angles in circles: individual questionsThe maths.scot topic page lists each National 5 angles-in-circles question separately (2015 to 2024).maths.scot Open ↗ (opens in a new tab)