Circle geometry

This section covers circles and circle geometry. This includes the parts of a circle, and all the circle theorems required for GCSE Maths or equivalent exams.

Tangent and Radius

Discover why a tangent meets a circle's radius at a right angle.

A tangent is a straight line that touches a circle at exactly one point. At the point of contact, the tangent is perpendicular to the radius drawn to that point.

This means the angle between the tangent and the radius is always 90°.

You will learn how to recognise this relationship in circle diagrams and use it to calculate missing angles. It is also useful when solving problems involving right-angled triangles formed by radii, tangents and other lines.

Understanding this theorem provides a valuable link between circle geometry and the angle rules you have already learned.

Two Radii

Use equal radii to reveal an isosceles triangle inside a circle.

A radius is a line segment from the centre of a circle to its circumference. All radii of the same circle have equal lengths.

When two radii are drawn to different points on the circumference, they form an isosceles triangle because two of its sides are equal.

You can use the properties of isosceles triangles to work out missing angles and understand the relationship between the centre of a circle and points on its circumference.

This simple but powerful idea often provides the first step in solving more complicated circle geometry problems.

Chord Bisector

Explore the relationship between a chord and the perpendicular line through its midpoint.

A chord is a straight line segment joining two points on a circle. Some chords have useful properties that allow you to identify equal lengths and right angles.

You will investigate the theorem that a perpendicular line from the centre of a circle to a chord bisects the chord, dividing it into two equal parts. Equivalently, the perpendicular bisector of a chord passes through the centre of the circle.

This relationship can help you calculate unknown lengths, recognise right-angled triangles within circles and solve problems involving radii and chords.

Angle at the Centre

Discover why the angle at the centre of a circle is twice the angle at the circumference.

When two points on a circle are joined to the centre and to another point on the circumference, they create angles with a special relationship.

The angle at the centre is twice the angle at the circumference when both angles subtend the same arc.

This theorem is one of the most useful results in circle geometry. You will learn how to identify the relevant angles in a diagram and use the relationship to calculate missing angles.

It also helps explain why different angles around the same circle can be linked, even when the diagram contains several intersecting lines.

Angle in a Semicircle

Learn why a triangle drawn inside a semicircle always contains a right angle.

When the diameter of a circle forms one side of a triangle and the third vertex lies on the circumference, the angle opposite the diameter is always 90°.

This is known as the angle-in-a-semicircle theorem.

You will learn how to recognise this pattern and use it to identify right-angled triangles hidden within circle diagrams. Once you have found the right angle, you can combine the theorem with the angle rules for triangles and, where appropriate, Pythagoras' theorem.

This is a useful example of how circle geometry connects with other areas of GCSE Maths.

The circle theorems are a set of rules of circle geometry that allow us to find sides and angles related to circles in various different situations. The GCSE curriculum, at the time of writing, covers seven circle theorems:

• Perpendicular bisector of a chord
• Angle at the centre of a circle is twice the angle at the circumference
• Angle in a semicircle is 90°
• Angles in the same segment of a circle are equal
• Opposite angles in a cyclic quadrilateral add up to 180°
• The two tangent theorems:
  • Tangent and radius of a circle meet at 90°
  • Two tangents from a point have equal length
• Alternate segment theorem

As you can see, there are actually two tangent theorems, but in GCSE Maths we count these as two parts of the same theorem.

There is another useful fact about circles, although this is not usually classed as a circle theorem:

• Two radii form an isosceles triangle

You might see some websites that say there are eight or nine circle theorems. That will usually be because they are counting both tangent theorems, and/or the two radii rule.

These resources are designed to support GCSE Maths students. They cover a range of useful mathematical topics but are not intended to provide a complete GCSE course or follow any particular exam board's specification. More...

Related articles

• Parts of a circle
• Perpendicular bisector of chord theorem
• Angle at the centre of a circle is twice the angle at the circumference
• Angle in a semicircle is 90°
• Angles in the same segment of a circle are equal
• Opposite angles in a cyclic quadrilateral add up to 180°
• Tangent and radius of a circle meet at 90°
• Two tangents from a point have equal length
• Alternate segment theorem
• Two radii form an isosceles triangle