Circle theorems for GCSE

Circle theorems are the rules for angles and lengths in a circle. GCSE Maths uses eight: the angle at the centre is twice the angle at the circumference, 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°, tangents from one point are equal, the perpendicular from the centre bisects a chord, and the alternate segment theorem.

The words you need

Circle theorem questions use the names of the parts of a circle, so learn these first. AQA lists them for GCSE: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment.

More on the basics in the diameter guide. And watch for radii: any two radii of the same circle are equal, so a triangle with two radii as sides is isosceles. Most circle theorem questions use that as well.

All circle theorems for GCSE

Here are all eight, with the reason to write in an exam. The wording follows AQA's specification, which lists these as the standard circle theorems students should apply and prove.

1. The angle at the centre is twice the angle at the circumference

Lines from the ends of the same arc meet at the centre O and at a point C on the circumference. The angle at O is always double the angle at C, as long as C is on the opposite side (the major arc). Here it is 140° and 70°. Going the other way, halve the centre angle to get the circumference angle.

2. The angle in a semicircle is 90°

This is theorem 1 when the arc is half the circle: the angle at the centre is a straight line, 180°, so the angle at the circumference is 90°. Any triangle drawn with a diameter as one side and its third corner on the circle has a right angle at that corner.

3. Angles in the same segment are equal

Angles at the circumference that stand on the same arc, and are on the same side of the chord, are equal. Both C and D here make 60° with A and B, because each is half of the same centre angle. Look for a "bow tie" or "fish" shape of two triangles on the same chord.

4. Opposite angles of a cyclic quadrilateral add up to 180°

In a quadrilateral with all four corners on the circle, opposite angles are supplementary. Here 95° + 85° = 180° and 93° + 87° = 180°. Check every corner is on the circle first: if one is inside or outside, the rule doesn't apply.

5. A tangent is perpendicular to the radius

Where a tangent touches the circle, it meets the radius at exactly 90°. This often creates a right-angled triangle, so Pythagoras' theorem or trigonometry can come next.

6. Tangents from an external point are equal

From a point P outside the circle you can draw exactly two tangents, and they are the same length: PA = PB. Together with the radii, they make a kite OAPB with right angles at A and B, which is useful for angle questions too.

7. The perpendicular from the centre bisects a chord

A line from the centre that meets a chord at 90° cuts it into two equal halves: AM = MB. It works the other way too: the line from the centre to the midpoint of a chord is perpendicular to it.

8. The alternate segment theorem

The angle between a tangent and a chord equals the angle in the alternate segment: the angle at the circumference on the other side of the chord. Here the tangent and chord PA make 55°, and angle PBA is also 55°. This is usually the hardest to spot: look for a tangent with a triangle drawn inside the circle from its point of contact.

How to answer circle theorem questions

1. Mark everything you know on the diagram, including equal radii and right angles at tangents.
2. Look for the shapes that trigger each theorem: a triangle on a diameter, a quadrilateral with every corner on the circle, a tangent touching a radius.
3. Find one angle at a time, using basic angle facts as well as circle theorems.
4. Write a reason for every angle, using the theorem's proper wording.

Worked examples

Example 1: centre angle and an isosceles triangle

O is the centre. A, B and C are on the circle, and angle AOB = 124°, with C on the major arc. Find angle ACB and angle OAB.

• Angle ACB = 124 ÷ 2 = 62°. Reason: the angle at the centre is twice the angle at the circumference.
• OA and OB are both radii, so triangle OAB is isosceles. Angle OAB = (180 − 124) ÷ 2 = 28°. Reason: base angles of an isosceles triangle are equal, and angles in a triangle add up to 180°.

Example 2: a tangent and a radius

A tangent touches a circle with centre O at T. A point P on the tangent is joined to O, and angle TPO = 34°. Find angle TOP.

Angle OTP = 90°, because a tangent is perpendicular to the radius at the point of contact. The angles in triangle OTP add up to 180°, so angle TOP = 180 − 90 − 34 = 56°.

Example 3: a cyclic quadrilateral with algebra

Two opposite angles of a cyclic quadrilateral are (3x + 10)° and (2x − 5)°. Find x and both angles.

So x = 35, and the angles are 3(35) + 10 = 115° and 2(35) − 5 = 65°. Reason: opposite angles of a cyclic quadrilateral add up to 180°.

Common mistakes

• Doubling instead of halving. The centre angle is the bigger one: centre = 2 × circumference.
• Using the cyclic quadrilateral rule on the wrong shape. All four corners must be on the circle. A quadrilateral with a corner at the centre is not cyclic.
• Missing the isosceles triangle. Two radii make two equal sides, so the base angles are equal.
• Vague reasons. "Circle theorem" or "tangent rule" won't get the mark. Write the full statement.
• Assuming a line is a diameter or a tangent. Only use it if the question says so or the diagram shows it through the centre.

To practise circle theorems among other topics in a short timed paper, try today's GCSE Maths Higher paper. Geometry comes up in most papers, but each one mixes topics.

Read next

Maths · GraphsHow to find the centre of a circle: 3 methodsHow to find the centre of a circle: by construction with two chords, from the equation by completing the square, and from three points, with examples.