What are the 7 Circle Theorems? (List with Diagrams & Examples)
The concept of circle theorems plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. These theorems show how angles, chords, tangents, and segments work inside a circle. Mastering circle theorems is useful for quick answering of geometry questions, especially in Class 9 and Class 10 exams. Let’s explore their meanings, key formulas, common mistakes, and simple tricks to remember them for your exams.
What Is Circle Theorem?
A circle theorem is a rule that describes the relationship between angles, lines, and points in a circle. You’ll find this concept applied in topics like chord geometry, tangent properties, and cyclic quadrilaterals. Knowing circle theorems lets you find unknown angles or prove shapes in a circle easily, without using a protractor.
List of Key Circle Theorems
- Angle at the Centre Theorem: The angle at the centre is twice the angle at the circumference subtended by the same arc.
- Angles in the Same Segment: Angles subtended by the same chord in the same segment are equal.
- Angle in a Semicircle: An angle formed in a semicircle is 90° (right angle).
- Perpendicular from Centre to Chord: The perpendicular from the centre to a chord bisects the chord.
- Equal Chords Theorem: Equal chords are equidistant from the centre.
- Tangent-Radius Theorem: The tangent to a circle is perpendicular to the radius at the point of contact.
- Cyclic Quadrilateral Theorem: Opposite angles of a cyclic quadrilateral sum to 180°.
- Alternate Segment Theorem: The angle between a tangent and a chord is equal to the angle in the alternate segment.
- Tangent-Secant Theorem: The square of the tangent equals the product of the external and the entire length of the secant.
Key Formula for Circle Theorems
Here’s the standard formula for the angle at the centre theorem: \( \text{Angle at the centre} = 2 \times \text{Angle at the circumference} \)
For the tangent-secant theorem: \( PT^2 = PQ \times PR \) where PT is the tangent, and PQ, PR are parts of the secant.
Step-by-Step Illustration
- You are given a circle with centre O. Chord AB subtends angle ACB at the circumference. Given ∠ACB = 40°.
- By the angle at centre theorem, the angle subtended by AB at the centre (∠AOB) is:
∠AOB = 2 × ∠ACB = 2 × 40° = 80°
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for circle theorem questions: If two chords are the same length, they are the same distance from the centre—no need to calculate distance every time; just check lengths.
Example Trick: When you see two equal chords, immediately declare their distances from the centre are equal—saving time during exams.
Try These Yourself
- State the angle made by a diameter at the edge of a circle.
- Given two equal chords, what is their distance from the centre?
- If a tangent meets a radius at a point, what angle do they form?
- A cyclic quadrilateral has one angle 110°. What is its opposite angle?
Frequent Errors and Misunderstandings
- Confusing ‘angle at centre’ with ‘angle at circumference’—remember, centre angle is always larger!
- Forgetting that the tangent makes 90° with the radius, not with the chord.
- Mistaking a cyclic quadrilateral’s opposite angles as equal (they sum to 180°, not always equal).
Relation to Other Concepts
The idea of circle theorems connects closely with cyclic quadrilaterals and tangent properties. Mastering these theorems also helps in trigonometry, area calculations, and proofs involving parts of a circle. Understanding circle theorems is especially helpful before learning about angles in polygons in higher geometry.
Classroom Tip
A quick way to remember key circle theorems is using the phrase: “Centre is twice, Tangent is right, Same segment means equal, Opposite in cyclic adds to 180°.” Vedantu’s teachers often use diagrams and colour codes in class to help students visualize and retain circle theorems better.
We explored circle theorems—from definitions, formulas, examples, and common mistakes to tricks and concept links. Continue practicing with Vedantu to become confident in solving all types of circle geometry problems and ace your exams!
FAQs on Circle Theorem: Rules, Diagrams & Exam Questions Explained
1. What is the fundamental concept of a circle theorem?
A circle theorem is a rule in geometry that describes a relationship between different parts of a circle, such as its angles, chords, arcs, and tangents. The fundamental concept is that these relationships are consistent and provable, allowing us to find unknown angles and lengths without direct measurement.
2. What are the key circle theorems for Class 9 Maths according to the CBSE 2025-26 syllabus?
For the CBSE 2025-26 Class 9 Maths syllabus, the key circle theorems are:
Equal chords of a circle subtend equal angles at the centre.
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Angles in the same segment of a circle are equal.
The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
3. Explain the theorem connecting the angle at the centre and the angle at the circumference.
This crucial theorem states that the angle an arc creates at the centre of a circle is exactly twice the angle it creates at any point on the circumference (the remaining part of the circle). For example, if an arc subtends an angle of 120° at the centre, it will subtend an angle of 60° at any point on the major arc.
4. What does it mean for angles to be in the same segment, and why are they always equal?
Angles in the same segment are angles subtended by the same arc at different points on the circumference. They are always equal because they are all half of the same angle subtended by the arc at the centre. This follows directly from the theorem that the angle at the centre is double the angle at the circumference.
5. How is the 'angle in a semicircle is a right angle' theorem a special case of another theorem?
The 'angle in a semicircle is a right angle (90°)' theorem is a direct consequence of the 'angle at the centre' theorem. A semicircle is formed by a diameter, which is a straight line subtending an angle of 180° at the centre. According to the theorem, the angle at the circumference must be half of the angle at the centre. Therefore, the angle at the circumference is 180° / 2 = 90°.
6. What are the two main properties of a cyclic quadrilateral according to circle theorems?
A cyclic quadrilateral is a four-sided figure whose vertices all lie on the circumference of a circle. Its two main properties are:
The sum of opposite angles is 180° (e.g., ∠A + ∠C = 180° and ∠B + ∠D = 180°).
The exterior angle is equal to the interior opposite angle.
7. What is the difference between a chord and a secant, and why is this important for circle theorems?
A chord is a line segment whose endpoints both lie on the circle. A secant is a line that intersects the circle at two distinct points. The difference is critical because theorems about chords (like bisecting them from the centre) apply to the segment *inside* the circle, while secants are lines that extend infinitely and are used in different theorems, often introduced in higher classes.
8. How do circle theorems apply in real-world examples like architecture or satellite navigation?
Circle theorems have practical applications in various fields. For example:
In architecture, they are used to design arches, domes, and curved windows, ensuring structural stability.
In satellite navigation (GPS), principles of circles and triangulation (which uses geometric properties) help pinpoint exact locations.
In engineering, they are used to design gears, wheels, and other circular components where angle and length precision is vital.
9. What is a common misconception about the theorem for cyclic quadrilaterals?
A common misconception is forgetting to check if all four vertices of the quadrilateral are actually on the circumference before applying the angle rule. Students sometimes incorrectly apply the rule that opposite angles sum to 180° to a non-cyclic quadrilateral, leading to incorrect calculations.
10. If equal chords are drawn in a circle, what can be said about the angles they subtend at the centre?
According to a fundamental circle theorem, if two chords in a circle are equal in length, they will subtend equal angles at the centre of the circle. This theorem provides a direct link between the length of a chord and the angle it forms at the circle's origin, which is very useful for proving triangle congruency within circles.
11. Can multiple circle theorems be used to solve a single geometry problem?
Yes, absolutely. Complex geometry problems often require the combined application of several circle theorems. For instance, you might first use the 'angles in the same segment' theorem to find one angle, then use the 'cyclic quadrilateral' property to find another, and finally use the 'angle at the centre' theorem to find a third. Recognising how these rules interact is key to solving advanced problems.
