IGCSE Maths / Circle theorems

Parts of a circle and cyclic quadrilaterals

Edexcel International GCSE Maths (4MA1), chapter 17: Circle theorems, specification point 4.6b. Free lesson.

Read the revision notes · IGCSE Maths past papers

Learn the names for parts of a circle and how to recognise a cyclic quadrilateral. If all four corners of a quadrilateral lie on a circle, its opposite angles add up to 180°.

How to name lines and arcs on a circle

A chord is a straight line joining two points on a circle. A tangent touches the circle at one point and continues past it, while an arc is part of the circle’s edge. The number of points where each meets the circle helps you tell them apart. A chord meets the circle at two points. A diameter is the longest chord and passes through the centre. A tangent touches the circle at one point. There is a tangent at every point on the circle. An arc is the circle’s edge between two points. The shorter way round is the minor arc; the longer way round is the major arc. A diameter…

How to tell a sector from a segment

A sector and a segment are different parts of a circle. Look at what forms their boundaries to identify them. A sector is bounded by two radii and the arc between them. The centre is one of its corners. A segment is bounded by one chord and an arc. The centre is not one of its corners.

How to recognise a cyclic quadrilateral

A cyclic quadrilateral is a four-sided shape with all four corners on the circumference of the same circle. All four corners must be on the circle for the angle rule to apply. Check each corner, not just three of them. If one corner is inside the circle, the shape is not a cyclic quadrilateral for this rule. The reason to give is: “Opposite angles of a cyclic quadrilateral add up to 180°.” Naming the shape alone is not the full reason.

How to find an unknown opposite angle

Opposite corners are diagonally across the quadrilateral from one another. Pair those angles, then subtract the known angle from 180°. Check the angle labels carefully so you use the angle opposite the one you know.

More lessons in Circle theorems

• Tangent and radius, and two tangents from a point
• The angle at the centre is twice the angle at the circumference
• The angle in a semicircle
• Angles in the same segment
• Opposite angles of a cyclic quadrilateral
• The alternate segment theorem
• Using the alternate segment theorem in a reasoning chain
• Reasoning chains with circle theorems
• Multi-step circle theorem problems
• Intersecting chords inside a circle
• Intersecting chords from a point outside the circle