Circle Theorems – A Must-Know Geometry Topic in GCSE Maths
In this blog, we will walk through all of the circle theorems required for GCSE Maths, explaining what each one means and how it is used in exam-style questions. By the end, you should feel confident identifying the correct theorem and applying it efficiently in problem-solving situations.
Here at Think Academy, we also provide circle theorems worksheets designed around GCSE exam questions, with structured practice to help students reinforce these concepts and improve exam performance. Come to try our online GCSE maths course for FREE, and get all the GCSE maths worksheets.
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What are circle theorems?
Circle theorems are a collection of geometric rules that describe how angles behave within a circle. In GCSE Maths, they form a key part of the geometry syllabus and are frequently assessed in both straightforward and multi-step exam questions. Mastering circle theorems allows students to calculate unknown angles accurately using logical reasoning, rather than relying on protractors.
These theorems are especially important because they bring together several core geometry ideas, including angle relationships, properties of triangles, and symmetry. In exams, circle theorems are often tested alongside other angle rules, meaning students must recognise which theorem applies and justify each step clearly to secure full marks.
There are seven main circle theorems:
- Angle at the centre theorem
- Angles in the same segment theorem
- Angle in a semicircle theorem
- Cyclic quadrilateral theorem
- Alternate segment theorem
- Tangent circle theorem
- Chord circle theorem
It is essential to understand each of these circle theorems and be able to explain them clearly using a single, precise statement.
Circle Theorem 1: Angles at the Centre and Angles at the Circumference
The angle at the centre is twice the angle at the circumference.
The angle subtended at the centre of a circle is exactly double the angle subtended at the circumference of a circle by the same arc (or by the same chord).
This theorem can be interpreted in different ways, depending on where the point lies on the circumference. In the examples below, the position of the point has been adjusted to demonstrate how the same theorem can appear in a variety of circle diagrams.
Remember: The angle at the centre is always larger than the angle at the circumference.
Circle Theorem 2: Angles in the same segment are equal
It states that any two angles at the circumference of a circle that are formed from the same two points on the circumference are equal.
In the diagram above, there is a chord divides the circle into the major and minor segments. This Theorem can be applied to any point that is placed on the major arc.
If the point lies on the minor arc in the opposite segment, the angle formed would be different; however, all angles subtended from the minor arc itself are equal.
Circle Theorem 3: The angle in a semicircle is a right angle
An angle in a semicircle is formed by connecting the two endpoints of a diameter to a single point on the circumference using chords. The two chords will always make an 90 degree angle, even the meeting point is placed on another point on the circumference.
Circle Theorem 4: Opposite angles in a cyclic quadrilateral
Opposite angles in a cyclic quadrilateral add up to 180 degrees.
A cyclic quadrilateral is a quadrilateral drawn inside a circle, where all 4 of its vertices must touch the circumference of the circle.
This theorem also shows that the interior angles of a cyclic quadrilateral add up to 360 degrees, which is a property shared by all quadrilaterals.
Circle Theorem 5: Alternate segment theorem
The angle between a tangent and a chord is equal to the angle, at the circumference, subtended by the same chord in the alternate segment.
Circle Theorem 6: Tangent of a circle
A tangent to a circle is a straight line that touches the circle at exactly one point.
1. The angle between a tangent and the radius is 90 degrees
Or we can say, a tangent and a radius that meet are perpendicular to each other.
2. The tangents which meet at the same point are equal in length.
Two tangents meet the circle at two different points (\( A \) and \( C \)) and they intersect at point \( B \). If the points \( A \) and \( C \) are linked by a chord, \( BA \) and \( BC \) are the same length so \( \triangle ABC \) is an isosceles triangle.
Circle Theorem 7: The Perpendicular Bisector of a Chord
- The perpendicular bisector of a chord passes through the centre of the circle.
- The perpendicular from the centre of a circle bisects a chord (cuts the chord into two equal parts).
Worked Examples
1. Angle at the centre theorem
\( A \), \( B \), and \( C \) are points on a circle with center \( O \).
\( \angle ABC = 52^\circ \), Calculate the size of angle \( x \).
The angle at the centre is twice the angle at the circumference, and so as we know the angle at the circumference, we need to multiply this number by 2 to get the angle \( AOC \):
\[ \angle AOC = 52^\circ \times 2 \] \[ \angle AOC = 104^\circ \]
2. Angles in the same segment theorem
\( A \), \( B \), \( C \), and \( D \) are points on the circumference of a circle. \( \angle ABC = 52^\circ \).
Find the size of \( \angle ADC \).
The angle \( ADC \) is in the same segment as the angle \( ABC \) and so we can state:
\(\angle ADC = \angle ABC = 52^\circ \)
3. Angle in a semicircle theorem
\( \triangle ABC \) is a triangle where \( A \), \( B \), and \( C \) lie on the circumference of a circle, and \( AB \) is a diameter.
\( \angle ABC = 62^\circ \), Calculate the size of \( \angle BAC \).
As the angle in a semicircle is equal to \( 90^\circ \), \( \angle ACB = 90^\circ \). We can therefore use the fact that the angles in a triangle total \( 180^\circ \) to calculate the size of \( \angle BAC \):
\[ \angle BAC = 180^\circ – (90^\circ + 62^\circ) \] \[ \angle BAC = 28^\circ \]
4. Cyclic quadrilateral theorem
\( ABCD \) is a quadrilateral where \( A \), \( B \), \( C \), and \( D \) lie on the circumference of a circle.
\( \angle ABC = 52^\circ \), calculate the size of \(\angle ADC \).
As opposite angles in a cyclic quadrilateral add up to \( 180^\circ \), we can calculate the size of angle \( \angle ADC \):
\[ \angle ADC = 180^\circ – 52^\circ \] \[ \angle BCD = 128^\circ \]
5. Alternate segment theorem
The triangle \( ABC \) is inscribed in a circle with centre \( O \). The tangent \( DE \) meets the circle at the point \( A \).
\( \angle CAE = 52^\circ \), calculate the size of \(\angle ABC \).
From the alternate segment theorem:
The angle between the tangent and the chord (\( \angle CAE \)) is equal to the angle subtended by the associated chord in the alternate segment (\( \angle ABC \)).
Therefore:
\[ \angle ABC = 52^\circ \]
6. Tangent circle theorem
Points \( A \), \( B \), and \( C \) are on the circumference of a circle with centre \( O \). \( DE \) is a tangent at point \( A \).
\( \angle ACB = 28^\circ \), calculate the size of \( \angle BAE \).
As \( AC \) is a diameter and the angle in a semicircle is \( 90^\circ \), \( \angle ABC = 90^\circ \), also the angles in a triangle total \( 180^\circ \).
\[ \angle CAB = 180^\circ – (90^\circ + 28^\circ) \]
\[ \angle CAB = 62^\circ \]
As the angle between the tangent and the radius is \( 90^\circ \), we can now calculate the angle \( BAE \):
\[ \angle BAE = 90^\circ – 62^\circ \] \[ \angle BAD = 28^\circ \]
7. Chord circle theorem
The diagram below shows a circle with centre, \( O \). Two points, \( A \) and \( B \), lie on its circumference.
The radius of the circle is \( 6 \, \text{cm} \). \( \angle OAB = 44^\circ \).
Find the length \( AB \). Give your answer correct to $3$ significant figures.
Label the radius on the diagram \( 6 \, \text{cm} \). Draw a perpendicular line from \( O \) to the line \( AB \), at point \( M \).
\( M \) will be the midpoint of the line \( AB \). The angle formed between the \( OM \) and \( PQ \) is a right angle.
Use SOHCAHTOA on triangle \( OAM \) to find the length \( AM \):
\[ \cos 44^\circ = \frac{AM}{6} \]
Rearranging: \[ AM = 6 \cos 44^\circ \]
\[ AM \approx 4.3160… \]
Double \( AM \) to find the length \( AB \): \[ AB = 2 \times 4.3160… \approx 8.6320… \]
Round \( AB \) to 3 significant figures: \[ AB = 8.63 \, \text{cm} \, (3 \, \text{s.f.}) \]
Common Misconceptions in Circle Theorems
- Confusing the angle at the centre with the angle at the circumference, and forgetting that the central angle is twice the size of the corresponding angle on the circle.
- Assuming an angle is a right angle without checking whether the line involved is actually a diameter of the circle.
- Applying the angles in the same segment theorem when the points lie on different arcs of the circle.
- Mixing up tangent properties with chord properties, particularly when identifying right angles at the point of contact.
- Forgetting that only opposite angles in a cyclic quadrilateral are supplementary, rather than all four angles.
- Failing to justify each step using the correct theorem, which can result in losing method marks even if the final angle is correct.
- Measuring angles directly from the diagram instead of using geometric reasoning, which is NOT accepted in GCSE exams.
Read more:
Area of a Triangle Using Trigonometry – for GCSE Maths
