Circle TheoremsEdexcel IGCSE Maths: Revision notes

Section 1

What happens when an angle is drawn in a semicircle?

If a triangle is drawn with one side as the diameter of a circle, and the third vertex anywhere else on the circumference, the angle at that third vertex is always $90^\circ$ .

This works because the angle at the centre ( $180^\circ$ , since the diameter is a straight line through the centre) is twice the angle at the circumference standing on the same arc.

Spot it by looking for a triangle where one side passes through the centre of the circle — that side must be the diameter.

Key termsdiametercircumferencesemicircle

Exam tip

Exam shorthand: 'angle in a semicircle = 90°'. If you see a triangle inscribed with one side as a diameter, mark the opposite angle 90° immediately.

Common mistake

Don't assume any chord through the middle of a circle is a diameter — check it actually passes through the centre point, not just looks central.

Section 2

How does the angle at the centre compare with the angle at the circumference?

When two points on a circle are joined to the centre and also joined to a third point on the circumference, both angles stand on the same arc. The angle at the centre is always exactly double the angle at the circumference.

So if the angle at the circumference is $x$ , the angle at the centre is $2x$ . Rearranged: angle at circumference $= \dfrac{1}{2} \times$ angle at centre.

This is the 'parent' theorem — the semicircle rule is really just the special case where the angle at the centre is $180^\circ$ .

Key termsangle at the centreangle at the circumferencesubtend

Example

If the angle at the circumference is $35^\circ$ , the angle at the centre subtended by the same arc is $70^\circ$ .

Common mistake

Only use this rule when both angles are subtended by the SAME arc — check both angles 'look at' the identical pair of points on the circle.

Section 3

What is true about angles in the same segment?

Angles subtended by the same arc, from different points on the same (major or minor) arc, are always equal to each other. This is because both angles are half of the same angle at the centre.

Look for two or more triangles sharing the same base chord, with their apex points on the same side of that chord — those apex angles are equal.

Key termssegmentchord

Think of it like this

Think of the chord as a stage and the equal angles as different seats in the same section of the audience — everyone in that section sees the same 'view angle' to the two ends of the stage.

Common mistake

Angles on opposite sides of the chord are NOT equal — they're only equal if the apex points are on the same arc (same side).

Section 4

What do opposite angles in a cyclic quadrilateral add up to?

A cyclic quadrilateral has all four vertices on the circumference of a circle. Its opposite angles are supplementary — they sum to $180^\circ$ .

So if one angle is $x$ , the angle diagonally opposite it is $180^\circ - x$ . Also, the exterior angle at a vertex equals the interior opposite angle.

Key termscyclic quadrilateralsupplementary anglesexterior angle

Example

If a cyclic quadrilateral has angles $80^\circ$ and $x$ opposite each other, then $x = 180^\circ - 80^\circ = 100^\circ$ .

Exam tip

Pair up ALL four vertices first and label opposite pairs before writing any equation — it's easy to pair adjacent angles by mistake.

Section 5

What are the key tangent properties?

A tangent touches a circle at exactly one point and is always perpendicular ( $90^\circ$ ) to the radius drawn to that point of contact.

Two tangents drawn from the same external point to a circle are equal in length, and the line from the external point to the centre bisects the angle between the two tangents (creating a kite shape with the two radii).

Key termstangentpoint of contactradius

Exam tip

Whenever you see a tangent, immediately draw and mark the radius to the point of contact as $90^\circ$ — this unlocks most tangent questions.

Common mistake

Don't confuse a tangent (touches once) with a chord (crosses through, joining two points on the circumference).

Section 6

What does the alternate segment theorem state?

The angle between a tangent and a chord drawn from the point of contact equals the angle in the alternate segment — that is, the angle subtended by the same chord in the segment on the other side.

To apply it: find the tangent-chord angle, then find the angle inside the triangle on the far side of the chord that subtends the same chord — they are equal.

Key termsalternate segment theorem

Example

If the tangent-chord angle is $50^\circ$ , the angle in the alternate segment (subtended by the same chord, on the far side) is also $50^\circ$ .

Common mistake

'Alternate' means the segment on the OTHER side of the chord from the tangent angle — not the nearer one.

Must Know

Angle in a semicircle = $90^\circ$ .
Angle at the centre = 2 × angle at the circumference (same arc).
Angles in the same segment (same arc) are equal.
Opposite angles in a cyclic quadrilateral sum to $180^\circ$ .
Tangent meets radius at $90^\circ$ ; tangents from the same external point are equal in length.
Alternate segment theorem: tangent-chord angle = angle in the alternate segment.

That's the notes covered.

Carry on to the next subtopic.