Circle Theorems – GCSE Maths

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Introduction

  • Circle Theorems are a set of rules and properties related to angles, chords, and segments in a circle.
  • They describe relationships between different geometric elements within and around a circle.

 

What are Circle Theorems?

  • Circle theorems are special rules in geometry that describe relationships between angles, lines, and arcs in a circle.
  • They help us find unknown angles or lengths using properties like angles in a semicircle, angles at the centre, and cyclic quadrilaterals, without the use of a protractor.
  • This has very useful applications in engineering and design for analyzing circular patterns and structures.
  • There are seven main circle theorems.

Basic Terminology of a Circle

  • Radius(r): A line from the center of the circle to any point on its edge.

Diagram showing the radius of a circle from the centre to a point on the circumference for GCSE Maths and maths revision.

  • Diameter(d): A line passing through the center, touching two points on the circle, equal to twice the radius.

Diagram showing the diameter of a circle passing through the centre for GCSE Maths and maths revision.

  • Circumference: The total distance around the circle.

Diagram highlighting the circumference of a circle for GCSE Maths geometry and maths revision.

  • Chord: A line joining any two points on the circle but does not have to pass through the center.

Diagram showing a chord inside a circle for GCSE Maths geometry and maths revision.

  • Tangent: A line that touches the circle at exactly one point and does not cross it.

Diagram showing a tangent touching a circle at one point for GCSE Maths geometry and maths revision.

  • Arc: A part of the circumference between two points.

Diagram showing an arc as part of the circumference of a circle for GCSE Maths and maths revision.

  • Sector: A part of a circle between two radii and the arc.

Diagram showing a sector of a circle formed by two radii and an arc for GCSE Maths and maths revision.

  • Segment: The area between a chord and the arc above it.

Diagram showing a segment of a circle formed by a chord and an arc for GCSE Maths and maths revision.

Circle Theorem 1 : The Alternate Segment

    • The angle that lies between a tangent and a chord is the same as the angle in the opposite part of the circle.
    • It helps to find unknown angles in circle problems easily when tangents and chords are involved in geometry questions.

Diagram explaining the Alternate Segment Theorem, which states that the angle formed between a tangent and a chord at the point of contact equals the angle in the opposite segment of the circle, used for GCSE Maths and maths revision.

Steps to use the alternate segment theorem:

  • Step#1: Find and mark the important parts on the circle.
  • Step#2: Use other angle rules to find one of the angles.
  • Step#3: Use the alternate segment theorem to find the other missing angle easily.

certified Physics and Maths tutorSolved Example:

Example: Triangle ABC is inscribed in a circle with centre O. A tangent DE touches the circle at point A. If the angle CAE = 56∘, calculate the size of the angle ABC.

Question diagram using the Alternate Segment Theorem with angle values given and theta to be found, supporting GCSE Maths circle geometry and maths revision.

Solution:

Step#1: Find and mark the important parts on the circle

  • Given:
  • The tangent DE touching the circle at A.
  • The chord AC meeting the tangent at A.
  • The angle CAE = 56∘ (angle between the tangent and chord).

Step#2: Use other angle rules to find one of the angles.

  • Since we already know,

Solved example applying the Alternate Segment Theorem to calculate theta using given angle values for GCSE Maths circle geometry and maths revision.

  • No additional angle facts are needed for this step.

Step#3: Use the alternate segment theorem.

  • The Alternate Segment Theorem directly tells us that the angle between the tangent and the chord is equal to the angle in the opposite segment.
  • Thus, 

Solved example applying the Alternate Segment Theorem to calculate theta using given angle values for GCSE Maths circle geometry and maths revision.

Circle Theorem 2 : Angles at the Centre and at the Circumference

  • The angle at the centre of a circle is twice the angle at the circumference when both angles stand on the same arc.
  • It helps to find unknown angles in circle geometry problems when we know one of the two angles.Diagram explaining the circle theorem of angles at the centre and circumference

Steps to use the angle at the center theorem:

  • Step#1: Find and mark the important parts on the circle
  • Step#2: Use other angle rules we know to find the angle at the centre or the angle at the edge (circumference).
  • Step#3: Use the angle at the centre theorem to find the missing angle

certified Physics and Maths tutorSolved Example:

Example: In a circle with centre C, A, B, and D lie on the circumference, and if ∠BCD = 150∘, find ∠BAD.solved example for understanding the circle theorem of angles at the center and the circumferenceSolution:

Step#1: Find and mark the important parts on the circle-

  • Given:
  • Angle at centre ∠BCD = 150∘
  • Angle at circumference ∠BAD = θ on the same arc.
  • We have radius BC and DC.
  • AB and AD are chords.

Step#2: Use other angle rules-

  • Since we already know,solved example for understanding the circle theorem of angles at the center and the circumference
  • No additional angle facts are needed for this step.

Step#3: Use the angle at the center theorem to find the missing angle-

  • Since the angle at the center is twice the angle at the circumference, we divide the given central angle by 2 to find ∠BAD.solved example for understanding the circle theorem of angles at the center and the circumference

Circle Theorem 3 : Angles in the Same Segment

  • Angles in the same segment of a circle are equal.
  • If we draw two angles on the circumference standing on the same chord, they will be equal, no matter where they are on that arc.
  • It helps us to find unknown angles in circle geometry problems when angles stand on the same chord.Diagram of a circle explaining the circle theorem of angles in the same segment

Steps to use the angles in the same segment theorem:

  • Step#1: Find and mark the important parts on the circle.
  • Step#2: Use any known angle rules to find one of the angles on the circumference in that segment.
  • Step#3: Use the angles in the same segment theorem to find the other angle (it will be equal).

certified Physics and Maths tutorSolved Example:

Example: In the circle below with centre O, if ∠DBC = 47∘, calculate the size of ∠CAD.diagram for solved example for circle theorem of angles in the same segment where we need to find the other angle when one is given

Solution:

Step#1: Find and mark the important parts on the circle-

Given:

  • The angle CBD = 47o
  • AC and BD are chords

Step#2: Use any known angle rules to find one of the angles on the circumference in that segment-

  • Since we already know,solution for solved example for circle theorem of angles in the same segment where we need to find the other angle when one is given
  • No additional angle facts are needed for this step.

Step#3: Use the angles in the same segment theorem to find the other angle (it will be equal)-

  • Using the Circle Theorem (angles in the same segment are equal):
  • Thus,solution for solved example for circle theorem of angles in the same segment where we need to find the other angle when one is given

Circle Theorem 4 : Angles in a Semicircle

  • The angle in a semicircle is always 90∘.
  • If we draw a triangle using the diameter of a circle, then the angle opposite the diameter will always be 90∘ or right angle.educational diagram explaining the circle theorem of angles in a semi circle for gcse maths

Steps to use the angles in a semicircle theorem:

  • Step#1: Find and mark the diameter and the triangle on the circle.
  • Step#2: Use known angle facts to find any other needed angles in the triangle if required.
  • Step#3: Use the semicircle theorem to state that the angle opposite the diameter is 90∘.

certified Physics and Maths tutorSolved Example:

Example: In a circle, ABC is a triangle with AB as the diameter and ∠ABC = 58∘. Find ∠BAC.educational diagram explaining the circle theorem of angles in a semi circle for gcse maths with a solved exampleSolution:

Step#1: Find and mark the diameter and the triangle on the circle-

  • Given:
  • AB is the diameter.
  • △ABC lies on the circle.

Step#2: Use known angle facts-

Sum of angles in a triangle:solution of solved example explaining the circle theorem of angles in a semi circle for gcse mathsStep#3: Use the semicircle theorem-

As the angle in a semicircle is equal to 90o, sosolution of solved example explaining the circle theorem of angles in a semi circle for gcse mathsThen, sum of angles:solution of solved example explaining the circle theorem of angles in a semi circle for gcse maths

Circle Theorem 5 : Chord of a Circle

    • When we draw a perpendicular line from the center of a circle to any chord, it neatly splits that chord into two equal parts.
    • It helps us to find unknown lengths in geometry problems and proves equal parts on either side of the chord.

educational diagram explaining the circle theorem of chord of a circle that states that When we draw a perpendicular line from the center of a circle to any chord, it neatly splits that chord into two equal parts for gcse maths

Steps to find missing lengths using chords:

  • Step#1: Mark the important parts (centre, chord, and the perpendicular from the centre to the chord).
  • Step#2: Use any known angle rules if we need to find missing angles in the triangle formed.
  • Step#3: Use Pythagoras’ theorem or trigonometry to find the missing length.

certified Physics and Maths tutorSolved Example:

Example: Calculate the length of chord BC, given that AE = 5 cm, ∠ADE = 65°, and AB ⊥ CD at E, with O as the centre of the circle.educational diagram of solved example explaining the circle theorem of chord of a circle that states that When we draw a perpendicular line from the center of a circle to any chord, it neatly splits that chord into two equal parts for gcse mathsSolution:

Step#1: Find and mark the diameter and the triangle on the circle.

  • Given:
  • O is the centre of the circle.
  • Chord BC is perpendicularly bisected by OE (since AB⊥CD at E, and O is the centre).
  • AE = 5 cm, ∠ADE=65∘

Step#2: Use any known angle rules if we need to find missing angles in the triangle formed.

  • Angles:

∠ABC = ∠ADE = 65° (angles in the same segment are equal).

  • Lengths:

Since the centre line BE is perpendicular to chord AD, it splits it evenly. So, BE = AE = 5 cm.

Step#3: Find Radius OB-

Using △ABE:solution of solved example explaining the circle theorem of chord of a circle that states that When we draw a perpendicular line from the center of a circle to any chord, it neatly splits that chord into two equal parts for gcse mathsStep#4: Find Chord BC-

  • Find half-chord (BE):

solution of solved example explaining the circle theorem of chord of a circle that states that When we draw a perpendicular line from the center of a circle to any chord, it neatly splits that chord into two equal parts for gcse maths

  • Double it for full chord (BC):solution of solved example explaining the circle theorem of chord of a circle that states that When we draw a perpendicular line from the center of a circle to any chord, it neatly splits that chord into two equal parts for gcse maths

Circle Theorem 6 : Tangent of a Circle

  • At the point where a tangent touches a circle, it forms a right angle (90°) with the radius drawn to that point.
  • This theorem helps calculate unknown angles and verify right angles in circle geometry problems.

educational diagram explaining he circle theorem of tangent of a circle for gcse maths

Steps to use the tangent of a circle theorems:

  • Step#1: Mark the important parts.
  • Step#2: Use any other angle facts you know to find missing angles near the tangent.
  • Step#3: Use the tangent theorem to find the missing angle.

certified Physics and Maths tutorSolved Example:

Example: Points A, B, and C lie on the circumference of a circle with centre O. Line DE is a tangent at point AA. If angle ACB = 63∘ , find angle BAD.

educational diagram of solved example explaining he circle theorem of tangent of a circle for gcse mathsSolution:

Step#1: Mark the important parts.

  • Given:
  • DE is a tangent to the circle at point A.
  • AC is a chord that meets the tangent.
  • ∠BAD = θ is the angle in the alternate segment.
  • ∠ACB = 63∘

Step#2: Use any other angle facts you know to find missing angles near the tangent.

  • ∠ACB = 63∘ is on the opposite side of chord AC from the tangent.

Step#3: Use the tangent theorem.

  • Using the Alternate Segment Theorem, the angle between the tangent and the chord equals the angle in the alternate segment. So:

Circle Theorem 7 : Cyclic Quadrilateral

  • In a quadrilateral with all corners on the circle, the opposite angles add up to 180∘.
  • If a 4-sided shape is inside a circle, then:

educational diagram explaining circle theorem of cyclic quadrilateral for gcse maths

Steps to use the cyclic quadrilateral theorem:

  • Step#1: Mark the key parts.
  • Step#2: Use any angle rules you know to find one of the opposite angles in the quadrilateral.
  • Step#3: Use the cyclic quadrilateral theorem to find the other missing angle.
  •  

certified Physics and Maths tutorSolved Example:

Example: ABCD is a cyclic quadrilateral where A, B, C, and D lie on the circumference of a circle. If angle DAB = 58°, calculate the size of angle BCD.educational diagram for solved example explaining circle theorem of cyclic quadrilateral for gcse mathsSolution:

Step#1: Mark the key parts.

  • Given:
  • The angle BAD = 51o
  • The angle BCD = θ

Step#2: Use any angle rules you know to find one of the opposite angles in the quadrilateral.

This is a cyclic quadrilateral, so opposite angles add up to 180°.

  • In cyclic quadrilaterals:

solution for solved example explaining circle theorem of cyclic quadrilateral for gcse mathsStep#3: Use the cyclic quadrilateral theorem to find the other missing angle.solution for solved example explaining circle theorem of cyclic quadrilateral for gcse maths

certified Physics and Maths tutorSolved Example:

Problem: Points A, B, C, and D lie on a circle with centre O. BD is the diameter, and AC is a chord perpendicular to the diameter at point E. If BE = 3 cm and ∠CDE = 40°, calculate the distance x, which is the length from C to E.solved example for circle theoremsSolution:

Step#1: Mark the key parts-

Given:

CE is perpendicular to BD (right angle at E)

  • Triangle CDE is right-angled at E
  • BE = 3 cm, ∠CDE = 40°

Step#2: Use angle facts-

  • In triangle CDE:
  • ∠CED = 90° (since CE ⊥ BD)
  • ∠CDE = 40° (given)
  • Use angle sum in triangle:
solved example for circle theorems

Step#3: Use tan to find x-

solved example for circle theorems

certified Physics and Maths tutorSolved Example:

Problem2: A circle with centre O has four points on the circumference: A, B, C, and D. Angle ∠CAD = 17°. Find the size of angle ∠CBD.solved example for circle theoremsStep#1: Understand the Figure-

  • Solution:
  • ∠CAD and ∠CBD are angles subtended by the same chord CD on opposite sides of the circle.

Step#2: Apply the Circle Theorem-

Angles in the same segment are equal.

That means:

solved example for circle theorems

Step#3: Conclude the answer-

Since ∠CAD = 17°,

  • Then:

solved example for circle theorems

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