GCSE Maths · Circle Theorems · Higher · Summary

GCSE Maths Circle Theorems (Higher): summary and key words

Part of the GCSE Maths circle theorems chapter (Higher). Chapter overview · whole chapter on one page

Chapter summary

Skill	Method
Circle words	Radius (centre to circle), diameter (chord through centre), chord (joins two points on the circle), arc, sector (two radii), segment (a chord), tangent (touches at one point)
Radii triangle	Two radii are equal, so triangle OAB is isosceles: base angles equal, angles add up to 180°
Angle in a semicircle	If AB is a diameter and C is on the circle then angle ACB = 90°
Angle at the centre	Centre angle = 2 × circumference angle on the same arc. Minor-arc point: use the reflex angle
Same segment	Angles made by the same chord with corners on the same side of it are equal
Cyclic quadrilateral	Opposite angles add up to 180°; exterior angle = interior opposite angle
Tangent and radius	The angle between a tangent and the radius at the point of contact is 90°; use Pythagoras for lengths
Two tangents	Tangents from an external point are equal; OATB is a kite; angle AOB + angle ATB = 180°
Alternate segment	Angle between a tangent and a chord = angle in the alternate segment (the opposite side of the chord)
Chords	The perpendicular from the centre bisects the chord; r² = d² + (half chord)²
Algebra with angles	Equal angles: expression = expression. Angles adding to 180° (or 90°): sum of expressions = 180. Solve, then substitute back
Proof	Use letters, give a reason for each line, finish with a conclusion

I can name and describe a chord, tangent, arc, sector and segment.
I can use the angle in a semicircle and explain why it is 90°.
I can use and prove that the angle at the centre is twice the angle at the circumference, including with a reflex angle.
I can use angles in the same segment and find the equal pairs in a diagram with crossing chords.
I can use opposite angles and exterior angles of a cyclic quadrilateral, and prove the theorem.
I can use a tangent being perpendicular to the radius, and tangents from a point being equal, with angles and Pythagoras.
I can use and prove the alternate segment theorem.
I can use that the perpendicular from the centre bisects a chord to find lengths and angles.
I can combine theorems with isosceles triangles and algebra, giving a reason for every step in standard wording.
I can write a clear proof or “show that” argument with letters, reasons and a conclusion.

Chord: A straight line that joins two points on a circle.
Diameter: A chord that passes through the centre. It is twice the radius.
Radius: A straight line from the centre of a circle to a point on the circle.
Arc: A part of the circumference. The shorter is the minor arc, the longer is the major arc.
Sector: The region between two radii and an arc.
Segment: The region between a chord and an arc.
Tangent: A straight line that touches a circle at exactly one point (the point of contact).
Circumference: The distance all the way round a circle, or the circle itself.
Cyclic quadrilateral: A four-sided shape with all four corners on a circle.
Alternate segment: The segment on the opposite side of a chord from a given angle.
Bisect: To cut exactly in half.
Subtend / stand on: An angle stands on an arc if its arms end at the two ends of that arc.
Reflex angle: An angle bigger than 180° and smaller than 360°.
Kite: A quadrilateral with two pairs of equal adjacent sides. OATB is a kite when TA and TB are tangents.
Converse: The statement the other way round: if the angle is 90°, then the line is a diameter.