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Circle theorems: worked examples and method
Higher · Grade 6 · Non-calculator
This page sets out the method for circle theorems in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Circle theorems topic page.
Key facts
The angle at the centre is twice the angle at the circumference (standing on the same arc).
The angle in a semicircle is 90∘.
Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral add up to 180∘.
A tangent meets a radius at 90∘; two tangents from the same external point are equal in length.
Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment. Also, the perpendicular from the centre to a chord bisects the chord.
The method, step by step
Method: Spot, state, solve
When you see a circle with marked points and one or two given angles, look for the shape each theorem makes before doing any arithmetic.
Mark the centre O and any radii — two radii make an isosceles triangle.
Look for a diameter (semicircle → 90∘), an angle at the centre and one at the circumference on the same arc, a four-sided shape with all corners on the circle, or a tangent.
Write the theorem in words next to the angle you find.
Use ordinary angle facts (angles in a triangle sum to 180∘, base angles of an isosceles triangle are equal) to finish.
Check the answer is sensible on the diagram (acute angles look acute).
Method: two-step diagrams
When one theorem does not reach the asked angle — find a stepping-stone angle first.
Mark every angle you can read off immediately (isosceles radii, angle in a semicircle, and so on).
Write a reason next to each one, using the accepted wording.
Look again: the angle you just found often unlocks the one in the question.
If the diagram has a cyclic quadrilateral, opposite angles sum to 180∘ — that is often the second step.
Stop when you have the asked angle and two written reasons.
How the lesson explains it
Half the centre
The angle at the centre is 80 degrees. The angle at the circumference is 40.
Double the edge
The angle at the circumference is 35 degrees. The angle at the centre is 70.
Semicircle
The angle in a semicircle is 90. This is the angle at the circumference when the two arc ends are the ends of a diameter.
Cyclic quad
Facing angles inside a cyclic quadrilateral sum to 180.
Tangent
The angle between a tangent and the radius is 90.
Name the fact
Say which theorem you used, then give the number of degrees.
The angle at the centre
The angle at the centre is twice the angle at the circumference when both stand on the same arc. An angle of 140^° at the centre gives 70^° at the circumference on that arc. If you are given the angle at the rim, double it to get the angle at the centre. The rim point must sit on the rest of the circumference, not on the arc itself. A point on the minor arc is looking at the major arc, so it uses half of the reflex angle at the centre.
A semicircle is the case where the chord through the centre is a diameter, so the angle at the centre is 180^°. Half of 180^° is 90^°. The angle in a semicircle is therefore a right angle. If AB is a diameter and C is on the circumference, angle ACB is 90^°. The right angle is at C, the point on the rim, not at A or B unless a second fact puts it there.
Method: 1. Halve the centre angle to get the angle at the circumference on the same arc. 2. A centre angle of 140^° gives a rim angle of 70^°. 3. A diameter makes a centre angle of 180^°, so the angle in a semicircle is 90^°.
Same segment and cyclic quadrilaterals
Angles in the same segment are equal. They stand on the same arc, so each is half of the same angle at the centre. If one of them is 47^°, every other angle in that same segment standing on that arc is 47^°. They do not have to belong to the same triangle. Mark the arc first, then mark each angle that faces that arc from the same side of the chord.
A cyclic quadrilateral has all four vertices on the circle. Opposite angles sum to 180^°. An angle of 76^° has opposite angle 104^°. Angles that share a side are next to each other, not opposite, so do not subtract those from 180^° for this theorem. The same idea is angles in opposite segments: two angles on opposite sides of a chord sum to 180^°. The four angles of the quadrilateral should still sum to 360^°.
Method: 1. Copy angles that stand on the same arc. 2. An angle of 47^° in a segment matches every other angle on that arc. 3. In a cyclic quadrilateral, opposite angles add to 180^°, so 76^° pairs with 104^°.
Tangents
A tangent touches the circle at one point. The radius to that point is perpendicular to the tangent, so the angle between them is 90^°. Write the reason as radius perpendicular to a tangent. Two tangents from the same outside point are equal in length. If one tangent from P is 9 cm, the other is 9 cm. The two triangles formed with the centre are congruent, so the line from the centre to P bisects the angle at the centre and the angle at P.
The alternate segment theorem joins a tangent to a chord. The angle between the tangent and the chord equals the angle in the alternate segment, on the other side of the chord. If that angle in the alternate segment is 35^°, the angle between the tangent and the chord is 35^°. Use this when a tangent and a chord meet and you can see an angle on the other side of the chord. An angle on the same side of the chord is not the alternate segment.
Method: 1. Put a right angle between a radius and the tangent at the point of contact. 2. Two tangents from the same outside point are equal, so both from P are 9cm. 3. The angle between a tangent and a chord equals the angle in the alternate segment.
Choosing the reason
Read the diagram for a clue before you calculate. A radius meeting a tangent gives 90^°. A diameter with the third vertex on the rim also gives 90^°, but the reason is the angle in a semicircle. An angle at the centre and an angle on the rim, on the same arc, means double or half. Four points on the circle mean a cyclic quadrilateral, so look at the opposite angles. A tangent and a chord mean you should look into the alternate segment.
Write the reason in words next to the arithmetic. Angle at the centre is twice the angle at the circumference is a complete reason. Opposite angles of a cyclic quadrilateral sum to 180^° is a complete reason. Check that both angles really use the same arc. Halving the wrong centre angle, or subtracting two angles that share a side, spoils the diagram.
Method: 1. A diameter with the third point on the rim gives 90^°, by the angle in a semicircle. 2. Halve a centre angle, or double a rim angle, only when both stand on the same arc. 3. Opposite angles of a cyclic quadrilateral add to 180^°.
Giving a reason the way the mark scheme expects
Circle theorem questions nearly always say "give a reason for each stage of your working" or "you must give reasons". An answer of 56^° with no reason earns almost nothing. Each line of working needs the angle, the number and a short reason that names the theorem. "Circle theorem" or "because it is a circle" is too vague to score.
Learn the standard reasons as short phrases: "the angle at the centre is twice the angle at the circumference", "angles in the same segment are equal", "opposite angles of a cyclic quadrilateral add up to 180^°", "the angle in a semicircle is 90^°", "the tangent is perpendicular to the radius" and "angles in a triangle add up to 180^°". Include "base angles of an isosceles triangle are equal" when two radii meet.
Method: 1. Write the angle you have found, for example angle AOB = 112^°. 2. Write a short reason that names the theorem on the same line. 3. Use the angle you have found as the starting point for the next line. 4. Finish with the angle the question asked for, boxed or clearly labelled.
Tip: Mark every angle you find on the diagram as you go. It stops you from losing track of the chain.
Radii make isosceles triangles
Any two radii of a circle are the same length, so the triangle they form with a chord is isosceles. If the centre angle AOB is 76^°, the other two angles of triangle OAB share 180^° - 76^° = 104^° equally, so each is 52^°. Exam questions use this step to connect a centre angle to a base angle that touches a tangent or a chord.
Combine it with the other theorems. A tangent makes 90^° with the radius, so an angle between a tangent and a chord can be found by subtracting from 90^°. The reason for the isosceles step is "the two radii are equal, so the base angles are equal". Say that clearly, because it is a separate mark.
Method: 1. Spot two radii in the diagram and mark them equal. 2. Write that the triangle is isosceles and quote the reason. 3. Subtract the apex angle from 180^° and halve it to find each base angle. 4. Carry the angle on to the next theorem in the chain.
Two tangents from one point
If two tangents PA and PB are drawn from a point P to a circle with centre O, then PA = PB, and the quadrilateral OAPB is a kite with two right angles at A and B. So the angle at O and the angle at P add up to 180^° because the four angles of the quadrilateral add up to 360^°.
For example, if angle APB = 48^° then angle AOB = 360^° - 90^° - 90^° - 48^° = 132^°. From there, triangle OAB is isosceles and each base angle is (180^° - 132^°) ÷ 2 = 24^°. Questions often ask for these base angles or for the angle between the chord and the tangent. The same tangent lengths give the symmetry of the kite, so the line OP bisects angle AOB and angle APB. This can help you find an angle quickly.
Method: 1. Mark the right angles where the tangents meet the radii. 2. Use the angles in a quadrilateral to find the angle at the centre. 3. Use the isosceles triangle OAB to find the base angles. 4. Quote a reason at each stage and check the angles add up correctly.
Proofs, algebra and "not cyclic" (Higher)
Higher questions include algebra. If opposite angles of a cyclic quadrilateral are (2x + 10)^° and (3x - 5)^°, then (2x + 10) + (3x - 5) = 180. Solve this to get x = 35, and then work out the angle that the question asks for. Check that both angles are positive and that the other pair also adds to 180^°. Algebra questions are usually worth four or five marks, so show every line.
A "prove" question may ask you to show that a quadrilateral is not cyclic. Work out the sum of one pair of opposite angles and show that it is not 180^°. A clear statement is needed: "The opposite angles do not add up to 180^°, so the quadrilateral is not cyclic". Always write the numbers and the conclusion.
Method: 1. Write the sum of a pair of opposite angles, with numbers or algebra. 2. For algebra, form and solve an equation, then substitute back. 3. For a proof of "not cyclic", show the sum is not 180^°. 4. Write a conclusion that quotes the theorem you used.
Remember
The angle at the circumference is half the angle at the centre.
The angle in a semicircle is 90 degrees.
Opposite angles in a cyclic quadrilateral add to 180 degrees.